Answers to Supplementary Exercises

Chapter 1 - General Probability

$\quad 5{,}184$
$\quad$ (a) $120$
$\quad$ (b) $110$
$\quad 6{,}151{,}600{,}000$
$\quad$ (a) $4{,}060$
$\quad$ (b) $360$
$\quad$ (c) $2{,}700$
$\quad$ (d) $1{,}000$
$\quad 3\,430$
$\quad 3\,480$
$\quad$ (a) $720$
$\quad$ (b) $72$
$\quad$ (c) $144$
$\quad 88{,}200$
$\quad$ (a) $36$
$\quad$ (b) $26$
$\quad 240$
$\quad$ (a) $6$
$\quad$ (b) $6$
$\quad$(c) $6$
$\quad$(d) $11$
$\quad$ (a) $\dfrac {7}{15}$
$\quad$ (b) $\dfrac {4}{15}$
$\quad$ (c) $\dfrac {2}{15}$
$\quad$ (d) $\dfrac {9}{15}$
$\quad$ (a) $\dfrac {1}{11}$ $\quad$ (b) $\dfrac {6}{11}$
$\quad$ (a) $18$
$\quad$ (b) $6$
$\quad$ (c) $9$ $\quad$ (d) $A\cap B=\{($taco, burrito, arroz c/ leche), (taco, enchilada, arroz c/ leche$)$, $($taco, chilaquiles, arroz c/ leche$)\}$ $\quad$ (e) $6$ $\quad$ (f) $A\cap B\cap C=\{($arroz c/ leche, taco, enchilada$)\}$
$\quad \dfrac{1}{2}$
$\quad 0.3\overline 3$
$\quad 0.53$
$\quad 0.6$
$\quad 0.205$
$\quad 52\%$
$\quad 0.35$
$\quad 0.26$
$\quad \dfrac{1}{3}$
$\quad$ (a) $\dfrac {1}{8}$
$\quad$ (b) $\dfrac {11}{48}$
$\quad$ (c) $\dfrac {6}{11}$
$\quad 0.6$
$\quad 0.004926$
$\quad 0.25$
$\quad 0.04\overline {33}$
$\quad 0.0893$
$\quad 0.1728$
$\quad 0.2922$
$\quad$ (a) $\dfrac{1}{2}$
$\quad$ (b) $\dfrac{1}{8}$

Chapter 2 - Univariate Distributions

$\quad$ (a) $\{0, 1, 2\}$
$\quad$ (b) $0.8\overline{33}$ $\quad$ (c) $0.3\overline{33}$ $\quad$ (d) $0.6$
$\quad$ (a) $\dfrac {1}{36}$
$\quad$ (b) $\dfrac {1}{3}$
$\quad$ (c) $\dfrac {1}{5}$
$\quad 0.0004$
$\quad 0.5131$
$\quad 2$
$\quad 60{,}000$
$\quad 0.9245$
$\quad 1.0750$
$\quad \dfrac{1}{3}$
$\quad 0.2007$
$\quad 0.2380$
$\quad 220$
$\quad 1.4285$
$\quad 0.04$
$\quad$ (a) $\dfrac {3}{64}$
$\quad$ (b) $\dfrac {1}{256}$
$\quad -2.872$
$\quad$ (a) $\dfrac {3}{2}$
$\quad$ (b) $\dfrac {3}{5}$ $\quad$ (c) $\dfrac {7}{16}$
$\quad \dfrac {71}{36}$
$\quad 0.35$
$\quad$ 6:08:29 am
$\quad 0.5134$
$\quad 0.4348$
$\quad \dfrac{1}{9}$
$\quad 93.06$
$\quad 10$
$\quad \dfrac{1}{4}$
$\quad 0.0317$
$\quad 0.2498$
$\quad 0.4167$
$\quad 4{,}176$
$\quad 0.0794$
$<quad 0.2498$
$\quad 1{,}692$
$\quad 3$
$\quad 120{,}000$
$\quad$
$\quad 999$
$\quad 500$
$\quad 0.9343$
$\quad 1.90$
$\quad 328$
$\quad 0.7275$$
$\quad 0.1241$
$\quad 4{,}176$
$\quad 0.45$
$\quad 998.72$
$\quad$
$\quad 985$

Chapter 3 - Multivariate Distributions

$\quad$ (a) $\dfrac{5}{12}$

$\quad$ (b) The PMFs are defined as follows:

$\quad \Pr[X=x]= \begin{cases} \dfrac{13}{24}, & x=0,\\ \dfrac{11}{24}, & x=1,\ 0, & \text{otherwise} \end{cases}$

$\quad \Pr[Y=y]= \begin{cases} \dfrac{7}{24}, & y=0,\\ \dfrac{5}{12}, & y=1,\\ 0, & \text{otherwise} \end{cases}$

$\quad$ (c) $\dfrac{6}{13}$

$\quad$ (d) No. For example, $\Pr[Y=1\mid X=0]\neq \Pr[Y=1]$.
$\quad$ (a) $0$
$\quad$ (b) $0$
$\quad 100$
$\quad$ (a) $6$
$\quad$ (b) $10$
$\quad 0.7333$
$\quad 0.4888$
$\quad$ (a) $0.25$

$\quad$ (b) $X$ and $Y$ are not independent. For example, $\mathrm{Pr}[X=0, Y=0\,]\neq \mathrm{Pr}[X=0\,]\cdot \mathrm{Pr}[Y=0\,]$

$\quad$ (c) $-0.25$
$\quad$ (a) $0.5$

$\quad$ (b) No
$\quad 0.24$
$\quad 0.576$
$\quad 0$
$\quad 11$
$\quad 0.9545$
$\quad 0.1875$
$\quad 0.416\overline{6}$
$\quad$
$\quad$
$\quad$
$\quad 103.025$
$\quad 0.4138$
$\quad$
$\quad 0.25$
$\quad 0.8\overline{8}$
$\quad 0.820$
$\quad 5$
$\quad 0.414$
$\quad 0.07639$
$\quad 0.275$
$\quad 0.8186$
$\quad 12$
$\quad 0.8413$