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$\mathrm{~N}$ | 1.31 | 0.405 | 0.405 | Calculate the force on an alpha particle passing a gold atomic nucleus at a distance of $0.00300 Å$. | quan |
||
2.13 | Angstrom | 4 | 4 | When an electron in a certain excited energy level in a one-dimensional box of length $2.00 Å$ makes a transition to the ground state, a photon of wavelength $8.79 \mathrm{~nm}$ is emitted. Find the quantum number of the initial state. | quan |
||
$10^{26}$ | 2.11 | 3 | 3 | For a macroscopic object of mass $1.0 \mathrm{~g}$ moving with speed $1.0 \mathrm{~cm} / \mathrm{s}$ in a one-dimensional box of length $1.0 \mathrm{~cm}$, find the quantum number $n$. | quan |
||
$\mathrm{~kJ} / \mathrm{mol}$ | 13.2 | 432.07 | 432.07 | For the $\mathrm{H}_2$ ground electronic state, $D_0=4.4781 \mathrm{eV}$. Find $\Delta H_0^{\circ}$ for $\mathrm{H}_2(g) \rightarrow 2 \mathrm{H}(g)$ in $\mathrm{kJ} / \mathrm{mol}$ | quan |
||
$\mathrm{kJ} / \mathrm{mol}$ | 15.39 | 0.14 | 0.14 | The contribution of molecular vibrations to the molar internal energy $U_{\mathrm{m}}$ of a gas of nonlinear $N$-atom molecules is (zero-point vibrational energy not included) $U_{\mathrm{m}, \mathrm{vib}}=R \sum_{s=1}^{3 N-6} \theta_s /\left(e^{\theta_s / T}-1\right)$, where $\theta_s \equiv h \nu_s / k$ and $\nu_s$ is the vibrational frequency of normal mode $s$. Calculate the contribution to $U_{\mathrm{m}, \text { vib }}$ at $25^{\circ} \mathrm{C}$ of a normal mode with wavenumber $\widetilde{v} \equiv v_s / c$ of $900 \mathrm{~cm}^{-1}$. | quan |
||
$10^{-23} \mathrm{~J} / \mathrm{T}$ | 10.17 | 1.61 | 1.61 | Calculate the magnitude of the spin magnetic moment of an electron. | quan |
||
$\mathrm{eV}$ | 14.29 | screenshot answer is weird | $3 rac{1}{3}$ | 3.333333333 | A particle is subject to the potential energy $V=a x^4+b y^4+c z^4$. If its ground-state energy is $10 \mathrm{eV}$, calculate $\langle V\rangle$ for the ground state. | quan |
|
$\mathrm{~nm}$ | 2.27 | hint | 0.264 | 0.264 | For an electron in a certain rectangular well with a depth of $20.0 \mathrm{eV}$, the lowest energy level lies $3.00 \mathrm{eV}$ above the bottom of the well. Find the width of this well. Hint: Use $\tan \theta=\sin \theta / \cos \theta$ | quan |
|
7.56 | 0 | 0 | Calculate the uncertainty $\Delta L_z$ for the hydrogen-atom stationary state: $2 p_z$. | quan |
|||
1.6.1_b | 0.4908 | 0.4908 | A one-particle, one-dimensional system has $\Psi=a^{-1 / 2} e^{-|x| / a}$ at $t=0$, where $a=1.0000 \mathrm{~nm}$. At $t=0$, the particle's position is measured. Find the probability that the measured value is between $x=0$ and $x=2 \mathrm{~nm}$. |
$$
\begin{aligned}
\operatorname{Pr}(0 \leq x \leq 2 \mathrm{~nm}) & =\int_0^{2 \mathrm{~nm}}|\Psi|^2 d x=a^{-1} \int_0^{2 \mathrm{~nm}} e^{-2 x / a} d x \\
& =-\left.\frac{1}{2} e^{-2 x / a}\right|_0 ^{2 \mathrm{~nm}}=-\frac{1}{2}\left(e^{-4}-1\right)=0.4908
\end{aligned}
$$
| quan |
||
$\mathrm{eV}$ | 6.6.1 | -13.598 | -13.598 | Calculate the ground-state energy of the hydrogen atom using SI units and convert the result to electronvolts. |
The $\mathrm{H}$ atom ground-state energy with $n=1$ and $Z=1$ is $E=-\mu e^4 / 8 h^2 \varepsilon_0^2$. Use of equation $$
\mu_{\mathrm{H}}=\frac{m_e m_p}{m_e+m_p}=\frac{m_e}{1+m_e / m_p}=\frac{m_e}{1+0.000544617}=0.9994557 m_e
$$ for $\mu$ gives
$$
\begin{gathered}
E=-\frac{0.9994557\left(9.109383 \times 10^{-31} \mathrm{~kg}\right)\left(1.6021766 \times 10^{-19} \mathrm{C}\right)^4}{8\left(6.626070 \times 10^{-34} \mathrm{~J} \mathrm{~s}\right)^2\left(8.8541878 \times 10^{-12} \mathrm{C}^2 / \mathrm{N}-\mathrm{m}^2\right)^2} \frac{Z^2}{n^2} \\
E=-\left(2.178686 \times 10^{-18} \mathrm{~J}\right)\left(Z^2 / n^2\right)\left[(1 \mathrm{eV}) /\left(1.6021766 \times 10^{-19} \mathrm{~J}\right)\right]
\end{gathered}
$$
$$
E=-(13.598 \mathrm{eV})\left(Z^2 / n^2\right)=-13.598 \mathrm{eV}
$$
a number worth remembering. The minimum energy needed to ionize a ground-state hydrogen atom is $13.598 \mathrm{eV}$.
| quan |
|
6.6.3 | 0.323 | 0.323 | Find the probability that the electron in the ground-state $\mathrm{H}$ atom is less than a distance $a$ from the nucleus. |
We want the probability that the radial coordinate lies between 0 and $a$. This is found by taking the infinitesimal probability of being between $r$ and $r+d r$ and summing it over the range from 0 to $a$. This sum of infinitesimal quantities is the definite integral
$$
\begin{aligned}
\int_0^a R_{n l}^2 r^2 d r & =\frac{4}{a^3} \int_0^a e^{-2 r / a} r^2 d r=\left.\frac{4}{a^3} e^{-2 r / a}\left(-\frac{r^2 a}{2}-\frac{2 r a^2}{4}-\frac{2 a^3}{8}\right)\right|_0 ^a \\
& =4\left[e^{-2}(-5 / 4)-(-1 / 4)\right]=0.323
\end{aligned}
$$
| quan |
||
$10^{-6}$ | 1.6.1_a | 4.979 | 4.979 | A one-particle, one-dimensional system has $\Psi=a^{-1 / 2} e^{-|x| / a}$ at $t=0$, where $a=1.0000 \mathrm{~nm}$. At $t=0$, the particle's position is measured. Find the probability that the measured value lies between $x=1.5000 \mathrm{~nm}$ and $x=1.5001 \mathrm{~nm}$. |
In this tiny interval, $x$ changes by only $0.0001 \mathrm{~nm}$, and $\Psi$ goes from $e^{-1.5000} \mathrm{~nm}^{-1 / 2}=0.22313 \mathrm{~nm}^{-1 / 2}$ to $e^{-1.5001} \mathrm{~nm}^{-1 / 2}=0.22311 \mathrm{~nm}^{-1 / 2}$, so $\Psi$ is nearly constant in this interval, and it is a very good approximation to consider this interval as infinitesimal. The desired probability is given as
$$
\begin{aligned}
|\Psi|^2 d x=a^{-1} e^{-2|x| / a} d x & =(1 \mathrm{~nm})^{-1} e^{-2(1.5 \mathrm{~nm}) /(1 \mathrm{~nm})}(0.0001 \mathrm{~nm}) \\
& =4.979 \times 10^{-6}
\end{aligned}
$$
| quan |
|
$\mathrm{~kJ}$ | 2.6 | -1.78 | -1.78 | In this example, $2.50 \mathrm{~mol}$ of an ideal gas with $C_{V, m}=12.47 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$ is expanded adiabatically against a constant external pressure of 1.00 bar. The initial temperature and pressure of the gas are $325 \mathrm{~K}$ and $2.50 \mathrm{bar}$, respectively. The final pressure is 1.25 bar. Calculate the final $ \Delta U$. | Because the process is adiabatic, $q=0$, and $\Delta U=w$. Therefore,
$$
\Delta U=n C_{\mathrm{v}, m}\left(T_f-T_i\right)=-P_{e x t e r n a l}\left(V_f-V_i\right)
$$
Using the ideal gas law,
$$
\begin{aligned}
& n C_{\mathrm{v}, m}\left(T_f-T_i\right)=-n R P_{\text {external }}\left(\frac{T_f}{P_f}-\frac{T_i}{P_i}\right) \\
& T_f\left(n C_{\mathrm{v}, m}+\frac{n R P_{\text {external }}}{P_f}\right)=T_i\left(n C_{\mathrm{v}, m}+\frac{n R P_{\text {external }}}{P_i}\right) \\
& T_f=T_i\left(\frac{C_{\mathrm{v}, m}+\frac{R P_{\text {external }}}{P_i}}{C_{\mathrm{v}, m}+\frac{R P_{\text {external }}}{P_f}}\right) \\
& =325 \mathrm{~K} \times\left(\frac{12.47 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}+\frac{8.314 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1} \times 1.00 \mathrm{bar}}{2.50 \mathrm{bar}}}{12.47 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}+\frac{8.314 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1} \times 1.00 \mathrm{bar}}{1.25 \mathrm{bar}}}\right)=268 \mathrm{~K} \\
&
\end{aligned}
$$
We calculate $\Delta U=w$ from
$$
\begin{aligned}
\Delta U & =n C_{V, m}\left(T_f-T_i\right)=2.5 \mathrm{~mol} \times 12.47 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1} \times(268 \mathrm{~K}-325 \mathrm{~K}) \\
& =-1.78 \mathrm{~kJ}
\end{aligned}
$$ | quan |
|
5.1 | $\frac{1}{\sqrt{4 \pi}}$ | 0.28209479 | Find $Y_l^m(\theta, \phi)$ for $l=0$. |
For $l=0$, Eq. $$
m=-l,-l+1,-l+2, \ldots,-1,0,1, \ldots, l-2, l-1, l
$$ gives $m=0$, and $$
S_{l, m}(\theta)=\sin ^{|m|} \theta \sum_{\substack{j=1,3, \ldots \ \text { or } j=0,2, \ldots}}^{l-|m|} a_j \cos ^j \theta
$$ becomes
$$
S_{0,0}(\theta)=a_0
$$
The normalization condition $$
\int_0^{\infty}|R|^2 r^2 d r=1, \quad \int_0^\pi|S|^2 \sin \theta d \theta=1, \quad \int_0^{2 \pi}|T|^2 d \phi=1
$$ gives$$\begin{aligned}
\int_0^\pi\left|a_0\right|^2 \sin \theta d \theta=1 & =2\left|a_0\right|^2 \\
\left|a_0\right| & =2^{-1 / 2}
\end{aligned}
$$
Equation $$
Y_l^m(\theta, \phi)=S_{l, m}(\theta) T(\phi)=\frac{1}{\sqrt{2 \pi}} S_{l, m}(\theta) e^{i m \phi}
$$ gives$$
Y_0^0(\theta, \phi)=\frac{1}{\sqrt{4 \pi}}
$$
| quan |
||
$10^{-10} \mathrm{~m}$ | 6.4 | 1.5377 | 1.5377 | The lowest-frequency pure-rotational absorption line of ${ }^{12} \mathrm{C}^{32} \mathrm{~S}$ occurs at $48991.0 \mathrm{MHz}$. Find the bond distance in ${ }^{12} \mathrm{C}^{32} \mathrm{~S}$. |
The lowest-frequency rotational absorption is the $J=0 \rightarrow 1$ line. $$h \nu=E_{\mathrm{upper}}-E_{\mathrm{lower}}=\frac{1(2) \hbar^2}{2 \mu d^2}-\frac{0(1) \hbar^2}{2 \mu d^2}
$$
which gives $d=\left(h / 4 \pi^2 \nu \mu\right)^{1 / 2}$. $$
\mu=\frac{m_1 m_2}{m_1+m_2}=\frac{12(31.97207)}{(12+31.97207)} \frac{1}{6.02214 \times 10^{23}} \mathrm{~g}=1.44885 \times 10^{-23} \mathrm{~g}
$$
The SI unit of mass is the kilogram, and
$$
\begin{aligned}
d=\frac{1}{2 \pi}\left(\frac{h}{\nu_{0 \rightarrow 1} \mu}\right)^{1 / 2} & =\frac{1}{2 \pi}\left[\frac{6.62607 \times 10^{-34} \mathrm{~J} \mathrm{~s}}{\left(48991.0 \times 10^6 \mathrm{~s}^{-1}\right)\left(1.44885 \times 10^{-26} \mathrm{~kg}\right)}\right]^{1 / 2} \\
& =1.5377 \times 10^{-10} \mathrm{~m}
\end{aligned}
$$
| quan |
|
$\mathrm{~N} / \mathrm{m}$ | 4.3 | 1855 | 1855 | The strongest infrared band of ${ }^{12} \mathrm{C}^{16} \mathrm{O}$ occurs at $\widetilde{\nu}=2143 \mathrm{~cm}^{-1}$. Find the force constant of ${ }^{12} \mathrm{C}^{16} \mathrm{O}$. |
The strongest infrared band corresponds to the $v=0 \rightarrow 1$ transition. We approximate the molecular vibration as that of a harmonic oscillator. The equilibrium molecular vibrational frequency is approximately
$$
\nu_e \approx \nu_{\text {light }}=\widetilde{\nu} c=\left(2143 \mathrm{~cm}^{-1}\right)\left(2.9979 \times 10^{10} \mathrm{~cm} / \mathrm{s}\right)=6.424 \times 10^{13} \mathrm{~s}^{-1}
$$
To relate $k$ to $\nu_e$, we need the reduced mass $\mu=m_1 m_2 /\left(m_1+m_2\right)$. One mole of ${ }^{12} \mathrm{C}$ has a mass of $12 \mathrm{~g}$ and contains Avogadro's number of atoms. Hence the mass of one atom of ${ }^{12} \mathrm{C}$ is $(12 \mathrm{~g}) /\left(6.02214 \times 10^{23}\right)$. The reduced mass and force constant are
$$
\begin{gathered}
\mu=\frac{12(15.9949) \mathrm{g}}{27.9949} \frac{1}{6.02214 \times 10^{23}}=1.1385 \times 10^{-23} \mathrm{~g} \\
k=4 \pi^2 \nu_e^2 \mu=4 \pi^2\left(6.424 \times 10^{13} \mathrm{~s}^{-1}\right)^2\left(1.1385 \times 10^{-26} \mathrm{~kg}\right)=1855 \mathrm{~N} / \mathrm{m}
\end{gathered}
$$
| quan |
|
5.8-5 (a) | $0.25$ | 0.25 | If the distribution of $Y$ is $b(n, 0.25)$, give a lower bound for $P(|Y / n-0.25|<0.05)$ when $n=100$. | stat |
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5.3-13 | $0.03$ | 0.03 | A device contains three components, each of which has a lifetime in hours with the pdf
$$
f(x)=\frac{2 x}{10^2} e^{-(x / 10)^2}, \quad 0 < x < \infty .
$$
The device fails with the failure of one of the components. Assuming independent lifetimes, what is the probability that the device fails in the first hour of its operation? HINT: $G(y)=P(Y \leq y)=1-P(Y>y)=1-P$ (all three $>y$ ). | stat |
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5.6-13 | $0.9522$ | 0.9522 | The tensile strength $X$ of paper, in pounds per square inch, has $\mu=30$ and $\sigma=3$. A random sample of size $n=100$ is taken from the distribution of tensile strengths. Compute the probability that the sample mean $\bar{X}$ is greater than 29.5 pounds per square inch. | stat |
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5.6-3 | $0.8185$ | 0.8185 | Let $\bar{X}$ be the mean of a random sample of size 36 from an exponential distribution with mean 3 . Approximate $P(2.5 \leq \bar{X} \leq 4)$ | stat |
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5.3-9 | $\frac{729}{4096}$ | 0.178 | Let $X_1, X_2$ be a random sample of size $n=2$ from a distribution with pdf $f(x)=3 x^2, 0 < x < 1$. Determine $P\left(\max X_i < 3 / 4\right)=P\left(X_1<3 / 4, X_2<3 / 4\right)$ | stat |
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7.4-1 | $117$ | 117 | Let $X$ equal the tarsus length for a male grackle. Assume that the distribution of $X$ is $N(\mu, 4.84)$. Find the sample size $n$ that is needed so that we are $95 \%$ confident that the maximum error of the estimate of $\mu$ is 0.4 . | stat |
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5.4-17 | $0.3359$ | 0.3359 | In a study concerning a new treatment of a certain disease, two groups of 25 participants in each were followed for five years. Those in one group took the old treatment and those in the other took the new treatment. The theoretical dropout rate for an individual was $50 \%$ in both groups over that 5 -year period. Let $X$ be the number that dropped out in the first group and $Y$ the number in the second group. Assuming independence where needed, give the sum that equals the probability that $Y \geq X+2$. HINT: What is the distribution of $Y-X+25$ ? | stat |
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9.6-11 | $9$ | 9 | Let $X$ and $Y$ have a bivariate normal distribution with correlation coefficient $\rho$. To test $H_0: \rho=0$ against $H_1: \rho \neq 0$, a random sample of $n$ pairs of observations is selected. Suppose that the sample correlation coefficient is $r=0.68$. Using a significance level of $\alpha=0.05$, find the smallest value of the sample size $n$ so that $H_0$ is rejected. | stat |
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7.3-5 | $0.1800$ | 0.1800 | In order to estimate the proportion, $p$, of a large class of college freshmen that had high school GPAs from 3.2 to 3.6 , inclusive, a sample of $n=50$ students was taken. It was found that $y=9$ students fell into this interval. Give a point estimate of $p$. | stat |
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7.4-15 | $144$ | 144 | If $\bar{X}$ and $\bar{Y}$ are the respective means of two independent random samples of the same size $n$, find $n$ if we want $\bar{x}-\bar{y} \pm 4$ to be a $90 \%$ confidence interval for $\mu_X-\mu_Y$. Assume that the standard deviations are known to be $\sigma_X=15$ and $\sigma_Y=25$. | stat |
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7.4-7 | $1068$ | 1068 | For a public opinion poll for a close presidential election, let $p$ denote the proportion of voters who favor candidate $A$. How large a sample should be taken if we want the maximum error of the estimate of $p$ to be equal to 0.03 with $95 \%$ confidence? | stat |
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5.5-15 (a) | $2.567$ | 2.567 | Let the distribution of $T$ be $t(17)$. Find $t_{0.01}(17)$. | stat |
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5.5-1 | $0.4772$ | 0.4772 | Let $X_1, X_2, \ldots, X_{16}$ be a random sample from a normal distribution $N(77,25)$. Compute $P(77<\bar{X}<79.5)$. | stat |
|||
5.4-19 | $0.5768$ | 0.5768 | 5.4-19. A doorman at a hotel is trying to get three taxicabs for three different couples. The arrival of empty cabs has an exponential distribution with mean 2 minutes. Assuming independence, what is the probability that the doorman will get all three couples taken care of within 6 minutes?
| stat |
|||
7.3-9 | $0.2115$ | 0.2115 | Consider the following two groups of women: Group 1 consists of women who spend less than $\$ 500$ annually on clothes; Group 2 comprises women who spend over $\$ 1000$ annually on clothes. Let $p_1$ and $p_2$ equal the proportions of women in these two groups, respectively, who believe that clothes are too expensive. If 1009 out of a random sample of 1230 women from group 1 and 207 out of a random sample 340 from group 2 believe that clothes are too expensive, Give a point estimate of $p_1-p_2$. | stat |
|||
5.6-9 | $0.6749$ | 0.6749 | Given below example: Approximate $P(39.75 \leq \bar{X} \leq 41.25)$, where $\bar{X}$ is the mean of a random sample of size 32 from a distribution with mean $\mu=40$ and variance $\sigma^2=8$. In the above example, compute $P(1.7 \leq Y \leq 3.2)$ with $n=4$ | stat |
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5.8-5 | $0.925$ | 0.925 | If the distribution of $Y$ is $b(n, 0.25)$, give a lower bound for $P(|Y / n-0.25|<0.05)$ when $n=1000$. | stat |
|||
7.5-1 | $0.7812$ | 0.7812 | Let $Y_1 < Y_2 < Y_3 < Y_4 < Y_5 < Y_6$ be the order statistics of a random sample of size $n=6$ from a distribution of the continuous type having $(100 p)$ th percentile $\pi_p$. Compute $P\left(Y_2 < \pi_{0.5} < Y_5\right)$. | stat |
|||
5.2-13 | 0.5117 | 0.5117 | Let $X_1, X_2$ be independent random variables representing lifetimes (in hours) of two key components of a
device that fails when and only when both components fail. Say each $X_i$ has an exponential distribution with mean 1000. Let $Y_1=\min \left(X_1, X_2\right)$ and $Y_2=\max \left(X_1, X_2\right)$, so that the space of $Y_1, Y_2$ is $ 0< y_1 < y_2 < \infty $ Find $G\left(y_1, y_2\right)=P\left(Y_1 \leq y_1, Y_2 \leq y_2\right)$. | stat |
|||
5.4-5 | $0.925$ | 0.925 | Let $Z_1, Z_2, \ldots, Z_7$ be a random sample from the standard normal distribution $N(0,1)$. Let $W=Z_1^2+Z_2^2+$ $\cdots+Z_7^2$. Find $P(1.69 < W < 14.07)$ | stat |
|||
5.3-1 | 0.0182 | 0.0182 | Let $X_1$ and $X_2$ be independent Poisson random variables with respective means $\lambda_1=2$ and $\lambda_2=3$. Find $P\left(X_1=3, X_2=5\right)$. HINT. Note that this event can occur if and only if $\left\{X_1=1, X_2=0\right\}$ or $\left\{X_1=0, X_2=1\right\}$. | stat |
|||
5.9-1 (a) | $0.9984$ | 0.9984 | Let $Y$ be the number of defectives in a box of 50 articles taken from the output of a machine. Each article is defective with probability 0.01 . Find the probability that $Y=0,1,2$, or 3 By using the binomial distribution. | stat |
|||
7.4-11 | $38$ | 38 | Some dentists were interested in studying the fusion of embryonic rat palates by a standard transplantation technique. When no treatment is used, the probability of fusion equals approximately 0.89 . The dentists would like to estimate $p$, the probability of fusion, when vitamin A is lacking. How large a sample $n$ of rat embryos is needed for $y / n \pm 0.10$ to be a $95 \%$ confidence interval for $p$ ? | stat |
|||
7.1-3 | $15.757$ | 15.757 | To determine the effect of $100 \%$ nitrate on the growth of pea plants, several specimens were planted and then watered with $100 \%$ nitrate every day. At the end of
two weeks, the plants were measured. Here are data on seven of them:
$$
\begin{array}{lllllll}
17.5 & 14.5 & 15.2 & 14.0 & 17.3 & 18.0 & 13.8
\end{array}
$$
Assume that these data are a random sample from a normal distribution $N\left(\mu, \sigma^2\right)$. Find the value of a point estimate of $\mu$. | stat |
|||
5.5-7 | $0.9830$ | 0.9830 | Suppose that the distribution of the weight of a prepackaged '1-pound bag' of carrots is $N\left(1.18,0.07^2\right)$ and the distribution of the weight of a prepackaged '3-pound bag' of carrots is $N\left(3.22,0.09^2\right)$. Selecting bags at random, find the probability that the sum of three 1-pound bags exceeds the weight of one 3-pound bag. HInT: First determine the distribution of $Y$, the sum of the three, and then compute $P(Y>W)$, where $W$ is the weight of the 3-pound bag. | stat |
|||
5.3-7 | $\frac{2}{5}$ | 0.4 | The distributions of incomes in two cities follow the two Pareto-type pdfs $$ f(x)=\frac{2}{x^3}, 1 < x < \infty , \text { and } g(y)= \frac{3}{y^4} , \quad 1 < y < \infty,$$ respectively. Here one unit represents $ 20,000$. One person with income is selected at random from each city. Let $X$ and $Y$ be their respective incomes. Compute $P(X < Y)$. | stat |
|||
7.3-3 | $0.5061$ | 0.5061 | Let $p$ equal the proportion of triathletes who suffered a training-related overuse injury during the past year. Out of 330 triathletes who responded to a survey, 167 indicated that they had suffered such an injury during the past year. Use these data to give a point estimate of $p$. | stat |
|||
6.1-1 | $1.1$ | 1.1 | One characteristic of a car's storage console that is checked by the manufacturer is the time in seconds that it takes for the lower storage compartment door to open completely. A random sample of size $n=5$ yielded the following times:
$\begin{array}{lllll}1.1 & 0.9 & 1.4 & 1.1 & 1.0\end{array}$ Find the sample mean, $\bar{x}$. | stat |
|||
5.5-1 (b) | $0.8561$ | 0.8561 | Let $X_1, X_2, \ldots, X_{16}$ be a random sample from a normal distribution $N(77,25)$. Compute $P(74.2<\bar{X}<78.4)$. | stat |
|||
5.3-3 | $\frac{36}{125}$
| 1.44 | Let $X_1$ and $X_2$ be independent random variables with probability density functions $f_1\left(x_1\right)=2 x_1, 0 < x_1 <1 $, and $f_2 \left(x_2\right) = 4x_2^3$ , $0 < x_2 < 1 $, respectively. Compute $P \left(0.5 < X_1 < 1\right.$ and $\left.0.4 < X_2 < 0.8\right)$. | stat |
|||
5.8-1 (a) | $0.84$ | 0.84 | If $X$ is a random variable with mean 33 and variance 16, use Chebyshev's inequality to find A lower bound for $P(23 < X < 43)$. | stat |
|||
6.3-5 | $0.2553$ | 0.2553 | Let $Y_1 < Y_2 < \cdots < Y_8$ be the order statistics of eight independent observations from a continuous-type distribution with 70 th percentile $\pi_{0.7}=27.3$. Determine $P\left(Y_7<27.3\right)$. | stat |
|||
5.4-21 | $0.4207$ | 0.4207 | Let $X$ and $Y$ be independent with distributions $N(5,16)$ and $N(6,9)$, respectively. Evaluate $P(X>Y)=$ $P(X-Y>0)$. | stat |
|||
7.4-5 | $257$ | 257 | A quality engineer wanted to be $98 \%$ confident that the maximum error of the estimate of the mean strength, $\mu$, of the left hinge on a vanity cover molded by a machine is 0.25 . A preliminary sample of size $n=32$ parts yielded a sample mean of $\bar{x}=35.68$ and a standard deviation of $s=1.723$. How large a sample is required? | stat |
|||
5.2-5 | 14.80 | 14.80 | Let the distribution of $W$ be $F(8,4)$. Find the following: $F_{0.01}(8,4)$. | stat |
|||
5.8-5 | $0.85$ | 0.85 | If the distribution of $Y$ is $b(n, 0.25)$, give a lower bound for $P(|Y / n-0.25|<0.05)$ when $n=500$. | stat |
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5.6-1 | $0.4772$ | 0.4772 | Let $\bar{X}$ be the mean of a random sample of size 12 from the uniform distribution on the interval $(0,1)$. Approximate $P(1 / 2 \leq \bar{X} \leq 2 / 3)$. | stat |
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5.2-9 | 840 | 840 | Determine the constant $c$ such that $f(x)= c x^3(1-x)^6$, $0 < x < 1$ is a pdf. | stat |
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5.3-15 | $0.0384$ | 0.0384 | Three drugs are being tested for use as the treatment of a certain disease. Let $p_1, p_2$, and $p_3$ represent the probabilities of success for the respective drugs. As three patients come in, each is given one of the drugs in a random order. After $n=10$ 'triples' and assuming independence, compute the probability that the maximum number of successes with one of the drugs exceeds eight if, in fact, $p_1=p_2=p_3=0.7$ | stat |
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5.2-11 | 0.1792 | 0.1792 | Evaluate
$$
\int_0^{0.4} \frac{\Gamma(7)}{\Gamma(4) \Gamma(3)} y^3(1-y)^2 d y
$$ Using integration. | stat |
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5.6-7 | $0.6247$ | 0.6247 | Let $X$ equal the maximal oxygen intake of a human on a treadmill, where the measurements are in milliliters of oxygen per minute per kilogram of weight. Assume that, for a particular population, the mean of $X$ is $\mu=$ 54.030 and the standard deviation is $\sigma=5.8$. Let $\bar{X}$ be the sample mean of a random sample of size $n=47$. Find $P(52.761 \leq \bar{X} \leq 54.453)$, approximately. | stat |
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5.3-19 | $5$ | 5 | Two components operate in parallel in a device, so the device fails when and only when both components fail. The lifetimes, $X_1$ and $X_2$, of the respective components are independent and identically distributed with an exponential distribution with $\theta=2$. The cost of operating the device is $Z=2 Y_1+Y_2$, where $Y_1=\min \left(X_1, X_2\right)$ and $Y_2=\max \left(X_1, X_2\right)$. Compute $E(Z)$. | stat |
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5.8-1 | $0.082$ | 0.082 | If $X$ is a random variable with mean 33 and variance 16, use Chebyshev's inequality to find An upper bound for $P(|X-33| \geq 14)$. | stat |
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5.5-9 (a) | $0.3085$ | 0.3085 | Suppose that the length of life in hours (say, $X$ ) of a light bulb manufactured by company $A$ is $N(800,14400)$ and the length of life in hours (say, $Y$ ) of a light bulb manufactured by company $B$ is $N(850,2500)$. One bulb is randomly selected from each company and is burned until 'death.' Find the probability that the length of life of the bulb from company $A$ exceeds the length of life of the bulb from company $B$ by at least 15 hours. | stat |
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Problem 1.4.15 | $\frac{1}{16}$ | 0.0625 | An urn contains 10 red and 10 white balls. The balls are drawn from the urn at random, one at a time. Find the probability that the fourth white ball is the fourth ball drawn if the sampling is done with replacement. | stat |
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Problem 1.4.5 | 0.9 | 0.9 | If $P(A)=0.8, P(B)=0.5$, and $P(A \cup B)=0.9$. What is $P(A \cap B)$? | stat |
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Problem 1.3.5 | $\frac{1}{3}$ | 0.33333333 | Suppose that the alleles for eye color for a certain male fruit fly are $(R, W)$ and the alleles for eye color for the mating female fruit fly are $(R, W)$, where $R$ and $W$ represent red and white, respectively. Their offspring receive one allele for eye color from each parent. Assume that each of the four possible outcomes has equal probability. If an offspring ends up with either two white alleles or one red and one white allele for eye color, its eyes will look white. Given that an offspring's eyes look white, what is the conditional probability that it has two white alleles for eye color? | stat |
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Problem 1.1.5 | $\frac{1}{6}$ | 0.166666666 | Consider the trial on which a 3 is first observed in successive rolls of a six-sided die. Let $A$ be the event that 3 is observed on the first trial. Let $B$ be the event that at least two trials are required to observe a 3 . Assuming that each side has probability $1 / 6$, find $P(A)$. | stat |
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Problem 1.3.9 | $1 - \frac{1}{e}$ | 0.6321205588 | An urn contains four balls numbered 1 through 4 . The balls are selected one at a time without replacement. A match occurs if the ball numbered $m$ is the $m$ th ball selected. Let the event $A_i$ denote a match on the $i$ th draw, $i=1,2,3,4$. Extend this exercise so that there are $n$ balls in the urn. What is the limit of this probability as $n$ increases without bound? | stat |
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Problem 1.1.1 | 0.68 | 0.68 | Of a group of patients having injuries, $28 \%$ visit both a physical therapist and a chiropractor and $8 \%$ visit neither. Say that the probability of visiting a physical therapist exceeds the probability of visiting a chiropractor by $16 \%$. What is the probability of a randomly selected person from this group visiting a physical therapist?
| stat |
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% | 1.5.3 | 15.1 | 15.1 | A doctor is concerned about the relationship between blood pressure and irregular heartbeats. Among her patients, she classifies blood pressures as high, normal, or low and heartbeats as regular or irregular and finds that 16\% have high blood pressure; (b) 19\% have low blood pressure; (c) $17 \%$ have an irregular heartbeat; (d) of those with an irregular heartbeat, $35 \%$ have high blood pressure; and (e) of those with normal blood pressure, $11 \%$ have an irregular heartbeat. What percentage of her patients have a regular heartbeat and low blood pressure? | stat |
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Problem 1.1.9 | $3(\frac{1}{3})-3(\frac{1}{3})^2+(\frac{1}{3})^3$ | 0.6296296296 | Roll a fair six-sided die three times. Let $A_1=$ $\{1$ or 2 on the first roll $\}, A_2=\{3$ or 4 on the second roll $\}$, and $A_3=\{5$ or 6 on the third roll $\}$. It is given that $P\left(A_i\right)=1 / 3, i=1,2,3 ; P\left(A_i \cap A_j\right)=(1 / 3)^2, i \neq j$; and $P\left(A_1 \cap A_2 \cap A_3\right)=(1 / 3)^3$. Use Theorem 1.1-6 to find $P\left(A_1 \cup A_2 \cup A_3\right)$. | stat |
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Problem 1.4.3 | $\frac{1}{6}$ | 0.166666666 | Let $A$ and $B$ be independent events with $P(A)=$ $1 / 4$ and $P(B)=2 / 3$. Compute $P(A \cap B)$ | stat |
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Problem 1.2.5 | 24 | 24 | How many four-letter code words are possible using the letters in IOWA if the letters may not be repeated? | stat |
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Problem 1.2.1 | 4096 | 4096 | A boy found a bicycle lock for which the combination was unknown. The correct combination is a four-digit number, $d_1 d_2 d_3 d_4$, where $d_i, i=1,2,3,4$, is selected from $1,2,3,4,5,6,7$, and 8 . How many different lock combinations are possible with such a lock? | stat |
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Problem 1.3.15 | 11 | 11 | An urn contains eight red and seven blue balls. A second urn contains an unknown number of red balls and nine blue balls. A ball is drawn from each urn at random, and the probability of getting two balls of the same color is $151 / 300$. How many red balls are in the second urn? | stat |
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Problem 1.1.1 | $\frac{2}{38}$ | 0.0526315789 | A typical roulette wheel used in a casino has 38 slots that are numbered $1,2,3, \ldots, 36,0,00$, respectively. The 0 and 00 slots are colored green. Half of the remaining slots are red and half are black. Also, half of the integers between 1 and 36 inclusive are odd, half are even, and 0 and 00 are defined to be neither odd nor even. A ball is rolled around the wheel and ends up in one of the slots; we assume that each slot has equal probability of $1 / 38$, and we are interested in the number of the slot into which the ball falls. Let $A=\{0,00\}$. Give the value of $P(A)$. | stat |
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Problem 1.3.13 | $\frac{8}{36}$ | 0.22222222 | In the gambling game "craps," a pair of dice is rolled and the outcome of the experiment is the sum of the points on the up sides of the six-sided dice. The bettor wins on the first roll if the sum is 7 or 11. The bettor loses on the first roll if the sum is 2,3 , or 12 . If the sum is $4,5,6$, 8,9 , or 10 , that number is called the bettor's "point." Once the point is established, the rule is as follows: If the bettor rolls a 7 before the point, the bettor loses; but if the point is rolled before a 7 , the bettor wins. Find the probability that the bettor wins on the first roll. That is, find the probability of rolling a 7 or 11 , $P(7$ or 11$)$. | stat |
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Problem 1.1.7 | 0.63 | 0.63 | Given that $P(A \cup B)=0.76$ and $P\left(A \cup B^{\prime}\right)=0.87$, find $P(A)$. | stat |
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Problem 1.2.3 | 6760000 | 6760000 | How many different license plates are possible if a state uses two letters followed by a four-digit integer (leading zeros are permissible and the letters and digits can be repeated)? | stat |
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Problem 1.4.1 | 0.14 | 0.14 | Let $A$ and $B$ be independent events with $P(A)=$ 0.7 and $P(B)=0.2$. Compute $P(A \cap B)$.
| stat |
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Problem 1.4.9 | 0.36 | 0.36 | Suppose that $A, B$, and $C$ are mutually independent events and that $P(A)=0.5, P(B)=0.8$, and $P(C)=$ 0.9 . Find the probabilities that all three events occur? | stat |
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Problem 1.2.17 | 0.00024 | 0.00024 | A poker hand is defined as drawing 5 cards at random without replacement from a deck of 52 playing cards. Find the probability of four of a kind (four cards of equal face value and one card of a different value). | stat |
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Problem 1.2.11 | 362880 | 362880 | Three students $(S)$ and six faculty members $(F)$ are on a panel discussing a new college policy. In how many different ways can the nine participants be lined up at a table in the front of the auditorium? | stat |
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Problem 1.4.17 | $1-(11 / 12)^{12}$ | 0.648004372 | Each of the 12 students in a class is given a fair 12 -sided die. In addition, each student is numbered from 1 to 12 . If the students roll their dice, what is the probability that there is at least one "match" (e.g., student 4 rolls a 4)? | stat |
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Problem 1.2.9 | 2 | 2 | The World Series in baseball continues until either the American League team or the National League team wins four games. How many different orders are possible (e.g., ANNAAA means the American League team wins in six games) if the series goes four games? | stat |
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Problem 1.1.3 | $\frac{12}{52}$ | 0.2307692308 | Draw one card at random from a standard deck of cards. The sample space $S$ is the collection of the 52 cards. Assume that the probability set function assigns $1 / 52$ to each of the 52 outcomes. Let
$$
\begin{aligned}
A & =\{x: x \text { is a jack, queen, or king }\}, \\
B & =\{x: x \text { is a } 9,10, \text { or jack and } x \text { is red }\}, \\
C & =\{x: x \text { is a club }\}, \\
D & =\{x: x \text { is a diamond, a heart, or a spade }\} .
\end{aligned}
$$
Find $P(A)$ | stat |
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Problem 1.3.7 | $\frac{1}{5}$ | 0.2 | An urn contains four colored balls: two orange and two blue. Two balls are selected at random without replacement, and you are told that at least one of them is orange. What is the probability that the other ball is also orange? | stat |
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Problem 1.5.1 | $\frac{21}{32}$ | 0.65625 | Bowl $B_1$ contains two white chips, bowl $B_2$ contains two red chips, bowl $B_3$ contains two white and two red chips, and bowl $B_4$ contains three white chips and one red chip. The probabilities of selecting bowl $B_1, B_2, B_3$, or $B_4$ are $1 / 2,1 / 4,1 / 8$, and $1 / 8$, respectively. A bowl is selected using these probabilities and a chip is then drawn at random. Find $P(W)$, the probability of drawing a white chip. | stat |
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Problem 1.1.13 | $\frac{2}{3}$ | 0.66666666666 | Divide a line segment into two parts by selecting a point at random. Use your intuition to assign a probability to the event that the longer segment is at least two times longer than the shorter segment. | stat |
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Problem 1.2.7 | 0.0024 | 0.0024 | In a state lottery, four digits are drawn at random one at a time with replacement from 0 to 9. Suppose that you win if any permutation of your selected integers is drawn. Give the probability of winning if you select $6,7,8,9$. | stat |
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Problem 1.4.19 | $ 1-1 / e$ | 0.6321205588 | Suppose that a fair $n$-sided die is rolled $n$ independent times. A match occurs if side $i$ is observed on the $i$ th trial, $i=1,2, \ldots, n$. Find the limit of this probability as $n$ increases without bound. | stat |
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Example 1.3.11 | 0.686 | 0.686 | An insurance company sells several types of insurance policies, including auto policies and homeowner policies. Let $A_1$ be those people with an auto policy only, $A_2$ those people with a homeowner policy only, and $A_3$ those people with both an auto and homeowner policy (but no other policies). For a person randomly selected from the company's policy holders, suppose that $P\left(A_1\right)=0.3, P\left(A_2\right)=0.2$, and $P\left(A_3\right)=0.2$. Further, let $B$ be the event that the person will renew at least one of these policies. Say from past experience that we assign the conditional probabilities $P\left(B \mid A_1\right)=0.6, P\left(B \mid A_2\right)=0.7$, and $P\left(B \mid A_3\right)=0.8$. Given that the person selected at random has an auto or homeowner policy, what is the conditional probability that the person will renew at least one of those policies? | The desired probability is
$$
\begin{aligned}
P\left(B \mid A_1 \cup A_2 \cup A_3\right) & =\frac{P\left(A_1 \cap B\right)+P\left(A_2 \cap B\right)+P\left(A_3 \cap B\right)}{P\left(A_1\right)+P\left(A_2\right)+P\left(A_3\right)} \\
& =\frac{(0.3)(0.6)+(0.2)(0.7)+(0.2)(0.8)}{0.3+0.2+0.2} \\
& =\frac{0.48}{0.70}=0.686 .
\end{aligned}
$$ | stat |
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Example 1.2.10 | 635013559600 | 635013559600 | What is the number of possible 13-card hands (in bridge) that can be selected from a deck of 52 playing cards? | The number of possible 13-card hands (in bridge) that can be selected from a deck of 52 playing cards is
$$
{ }_{52} C_{13}=\left(\begin{array}{l}
52 \\
13
\end{array}\right)=\frac{52 !}{13 ! 39 !}=635,013,559,600 .
$$
| stat |
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Example 1.2.5 | 5040 | 5040 | What is the number of ways of selecting a president, a vice president, a secretary, and a treasurer in a club consisting of 10 persons? | The number of ways of selecting a president, a vice president, a secretary, and a treasurer in a club consisting of 10 persons is
$$
{ }_{10} P_4=10 \cdot 9 \cdot 8 \cdot 7=\frac{10 !}{6 !}=5040 .
$$ | stat |
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Example 1.3.5 | $\frac{9}{24}$ | 0.375 | At a county fair carnival game there are 25 balloons on a board, of which 10 balloons 1.3-5 are yellow, 8 are red, and 7 are green. A player throws darts at the balloons to win a prize and randomly hits one of them. Given that the first balloon hit is yellow, what is the probability that the next balloon hit is also yellow? | Of the 24 remaining balloons, 9 are yellow, so a natural value to assign to this conditional probability is $9 / 24$. | stat |
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Example 1.2.8 | 311875200 | 311875200 | What is the number of ordered samples of 5 cards that can be drawn without replacement from a standard deck of 52 playing cards? |
The number of ordered samples of 5 cards that can be drawn without replacement from a standard deck of 52 playing cards is
$$
(52)(51)(50)(49)(48)=\frac{52 !}{47 !}=311,875,200 .
$$
| stat |
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Example 1.3.6 | $\frac{7}{30}$ | 0.23333333333 | A bowl contains seven blue chips and three red chips. Two chips are to be drawn successively at random and without replacement. We want to compute the probability that the first draw results in a red chip $(A)$ and the second draw results in a blue chip $(B)$. | It is reasonable to assign the following probabilities:
$$
P(A)=\frac{3}{10} \text { and } P(B \mid A)=\frac{7}{9} \text {. }
$$
The probability of obtaining red on the first draw and blue on the second draw is
$$
P(A \cap B)=\frac{3}{10} \cdot \frac{7}{9}=\frac{7}{30}
$$ | stat |
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Example 1.3.7 | 0.064 | 0.064 | From an ordinary deck of playing cards, cards are to be drawn successively at random and without replacement. What is the probability that the third spade appears on the sixth draw? | Let $A$ be the event of two spades in the first five cards drawn, and let $B$ be the event of a spade on the sixth draw. Thus, the probability that we wish to compute is $P(A \cap B)$. It is reasonable to take
$$
P(A)=\frac{\left(\begin{array}{c}
13 \\
2
\end{array}\right)\left(\begin{array}{c}
39 \\
3
\end{array}\right)}{\left(\begin{array}{c}
52 \\
5
\end{array}\right)}=0.274 \quad \text { and } \quad P(B \mid A)=\frac{11}{47}=0.234
$$
The desired probability, $P(A \cap B)$, is the product of those numbers:
$$
P(A \cap B)=(0.274)(0.234)=0.064
$$
| stat |
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Example 1.2.11 | 0.0000092 | 0.0000092 | What is the probability of drawing three kings and two queens when drawing a five-card hand from a deck of 52 playing cards? | Assume that each of the $\left(\begin{array}{c}52 \\ 5\end{array}\right)=2,598,960$ five-card hands drawn from a deck of 52 playing cards has the same probability of being selected.
Suppose now that the event $B$ is the set of outcomes in which exactly three cards are kings and exactly two cards are queens. We can select the three kings in any one of $\left(\begin{array}{l}4 \\ 3\end{array}\right)$ ways and the two queens in any one of $\left(\begin{array}{l}4 \\ 2\end{array}\right)$ ways. By the multiplication principle, the number of outcomes in $B$ is
$$
N(B)=\left(\begin{array}{l}
4 \\
3
\end{array}\right)\left(\begin{array}{l}
4 \\
2
\end{array}\right)\left(\begin{array}{c}
44 \\
0
\end{array}\right)
$$
where $\left(\begin{array}{c}44 \\ 0\end{array}\right)$ gives the number of ways in which 0 cards are selected out of the nonkings and nonqueens and of course is equal to 1 . Thus,
$$
P(B)=\frac{N(B)}{N(S)}=\frac{\left(\begin{array}{l}
4 \\
3
\end{array}\right)\left(\begin{array}{c}
4 \\
2
\end{array}\right)\left(\begin{array}{c}
44 \\
0
\end{array}\right)}{\left(\begin{array}{c}
52 \\
5
\end{array}\right)}=\frac{24}{2,598,960}=0.0000092 .
$$ | stat |
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Example 1.2.13 | 35 | 35 | In an orchid show, seven orchids are to be placed along one side of the greenhouse. There are four lavender orchids and three white orchids. How many ways are there to lineup these orchids? | Considering only the color of the orchids, we see that the number of lineups of the orchids is
$$
\left(\begin{array}{l}
7 \\
4
\end{array}\right)=\frac{7 !}{4 ! 3 !}=35 \text {. }
$$ | stat |
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Example 1.3.2 | 0.75 | 0.75 | If $P(A)=0.4, P(B)=0.5$, and $P(A \cap B)=0.3$, find $P(B \mid A)$. | $P(B \mid A)=P(A \cap B) / P(A)=0.3 / 0.4=0.75$. | stat |
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Example 1.2.9 | 2598960 | 2598960 | What is the number of possible 5-card hands (in 5-card poker) drawn from a deck of 52 playing cards? | The number of possible 5-card hands (in 5-card poker) drawn from a deck of 52 playing cards is
$$
{ }_{52} C_5=\left(\begin{array}{c}
52 \\
5
\end{array}\right)=\frac{52 !}{5 ! 47 !}=2,598,960
$$
| stat |