Answers to Selected Exercises

1.9. Yes, probably so. The outcomes are correct and incorrect and $p 1 4$ .

1.10. Yes, approximately. The outcomes are prefer $A$ and do not prefer $A$ ; $p$ is the proportion of voters in the entire district who prefer $A$ .

1.11. Yes, the outcomes are red and black, and $p 18 38$ .

1.12. No, probably not. The games are almost certainly dependent, and the win probably depends on who is serving and thus is not constant from game to game.

1.15. $r p p 2 p p 2 2 1 p 2 p 2 p 4$

1.22.

$k 10$ , $Y k 19.56$
$k 5$ , $Y k 426.22$
$k 40$ , $Y k 64.23$

2.29.

$X k 20 k 15 k 45 20 k$ for $k 01 20$
$X 4$
$X 16 5$
$X 12 0.000102$ . She has no hope of passing.

2.30.

$Y k 50 k 150 k 4950 50 k$ for $k 01 50$
$X 1$
$X 49 50$
$Y 3 0.9822$ .

2.31.

$Z k 50 k 25 k 35 50 k$ for $k 01 50$
$Z 20$
$Z 12$
$Z 19 0.3356$ .
$Z 19 0.3330$ .

2.32.

$N k 10 k 16 k 56 10 k$ for $k 01 10$
$N 5 3$
$N 25 18$

2.33. Let $Y n$ denote the number of heads in the first $n$ tosses.

Y 20 j Y 100 30 20 j 80 30 j 10030, j 01 20

2.34.

$Z k 1000 k 38 k 58 1000 k$ for $k 01 1000$
$Z 375$
$Z 1875 8$
$Z 400 0.0552$ .
$Z 400 0.0550$ .

2.36.

$5 Y 10 0.8815$
$5 Y 10 0.878$

2.37.

$0.5 M 0.7 0.8089$
$0.5 M 0.7 0.808$

2.39.

$at least 1 ace in 6 rolls 0.6651$
$at least 2 aces in 12 rolls 0.6187$

2.40.

$at least 1 ace in 4 rolls of 1 die 0.5177$
$at least 2 aces in 24 rolls of 2 dice 0.4914$

2.41. Proportion of females:

$m 0.433$
$m 0 0.636$
$m 1 0.259$
$m 2 0.5$

2.42. $m red 0.168$

2.48.

$r 32 p 3 p 2 2 p 3$ .
$r 53 p 10 p 3 15 p 4 6 p 5$ .
3 out of 5 is better for $p 1 2$

2.54.

$b 1 p 1 2 p$
$b 2 p 1 2 p$
$b 3 p 1 3 2 p 3 2 p 2 p 3$

3.16.

$U n 56 n 1 1 6, n$
$U 6$
$U 30$
$U 5 525 1296$

3.17.

$N n 4950 n 1 1 50, n$
$Z 50$
$Z 2450$
$N 20 0.6676$

3.18. 0.4

3.19. Geometric with $p 18 38$

3.24. $1000.

3.29. $W i 2 n i 2 n 1, i 12 n$

4.18.

$V n n 1 2 163 56 n 3, n 34$
$V 18$
$V 90$
$V 20 0.3643$

4.19.

$V 5 m V 10 25 m 1 4 24 m 4 249, m 56 20$
$V 5 V 10 25 25 2$
$V 5 V 10 25 375 44$

4.20.

$N n n 1 2 1504 4950 n 4, n 45$
$N 200$
$N 9800$
$N 200 0.5685$

4.21.

$8 V 5 15 0.7142$
$8 V 5 15 0.7445$

4.22. Let $V$ denote the number of tosses needed to get 50 heads.

0.0072
No.

4.38.

0.6825.
0.7102

4.44.

$f k k 1 3 1 2 k 1$ for $k 4567$ , $N 5.8125$ , $N 1.0136$ .
$f k k 1 3 0.7 4 0.3 k 4 0.3 4 0.7 k 4$ for $k 4567$ , $N 5.3780$ , $N 1.0497$ .
$f k k 1 3 0.9 4 0.1 k 4 0.1 4 0.9 k 4$ for $k 4567$ , $N 4.4394$ , $N 0.6831$ .

4.46. $A$ gets $72.56, $B$ gets $27.44

5.10.

0.0075
0.0178
0.205
0.123

5.11. $f u v w x y z 4 u v w x y z 14 u z 18 v w x y$ for $u$ , $v$ , $w$ , $x$ , $y$ , $z$ nonnegative integers that sum to 4

5.13.

$0.625$
$0.0386$

6.10.

0.7794
0.7752

6.18.

Probability density function of $M 5$ : $f 0 f 1 10 32$ , $f 2 f 3 5 32$ , $f 4 f 5 1 32$ .
$M 5 11 8$
$M 5 111 64$

6.19. $M 10 4 57 64$

6.32.

Probability density function of $L 10$ : $f 0 f 10 63 256$ , $f 2 f 8 35 256$ , $f 4 f 6 30 256$ ,
$L 10 5$
$L 10 15$

6.37. $3191 11895 0.2683$

6.41. $3 25$

6.45. $f 2 1 2$ , $f 4 1 8$ , $f 6 1 16$ , $f 8 5 128$ , $f 10 7 512$ .

Answers to Selected Exercises

11. Bernoulli Trials

1. Introduction

2. The Binomial Distribution

3. The Geometric Distribution

4. The Negative Binomial Distribution

5. The Multinomial Distribution

6. The Simple Random Walk