Chapter 7

$4\times <!--\; \times \; -->15=60$

$4\times <!--\; \times \; -->15\times <!--\; \times \; -->3\times <!--\; \times \; -->2=360$

$3\times <!--\; \times \; -->10\times <!--\; \times \; -->10\times <!--\; \times \; -->10\times <!--\; \times \; -->10\times <!--\; \times \; -->10\times <!--\; \times \; -->26\times <!--\; \times \; -->26\times <!--\; \times \; -->26=\text{5,272,800,000}$

120

720

132

70

30

24,024

5,814

${}_{15}{P}_3=\mathrm{2,730}$

combination

combination

15

45

2,002

30,856

111,930

290,004

$\{\mathrm{♡<!--\; ♡\; -->},\mathrm{♠<!--\; ♠\; -->},\mathrm{♣<!--\; ♣\; -->},\mathrm{♢<!--\; ♢\; -->}\}$

{A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K}

{1, 2, 3, 4}

{3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

Independent

Dependent

{H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}

{J$\mathrm{♡<!--\; ♡\; -->}$ Q$\mathrm{♡<!--\; ♡\; -->}$, J$\mathrm{♡<!--\; ♡\; -->}$ K$\mathrm{♡<!--\; ♡\; -->}$, Q$\mathrm{♡<!--\; ♡\; -->}$ J$\mathrm{♡<!--\; ♡\; -->}$, Q$\mathrm{♡<!--\; ♡\; -->}$ K$\mathrm{♡<!--\; ♡\; -->}$, K$\mathrm{♡<!--\; ♡\; -->}$ J$\mathrm{♡<!--\; ♡\; -->}$, K$\mathrm{♡<!--\; ♡\; -->}$ Q$\mathrm{♡<!--\; ♡\; -->}$}

$P(\text{H}\ge 5)=0$

$\frac{1}{6}$

$\frac{2}{3}$

$\frac{1}{2}$

$\frac{5}{16}$

$\frac{1}{3}$

$\frac{1}{2}$

$\frac{2}{3}$

$\frac{13}{1,000}=1.3\mathrm{\%<!--\; \%\; -->}$

theoretical

subjective

empirical

$\frac{13}{16}$

$\frac{144}{43680}=\frac{3}{910}$

$1-<!--\; -\; -->\frac{{}_{53}{C}_3}{{}_{58}{C}_3}\approx <!--\; \approx \; -->10.7\mathrm{\%<!--\; \%\; -->}$

$\frac{{}_{13}{C}_2\times <!--\; \times \; -->{}_{13}{C}_3}{{}_{52}{C}_5}\approx <!--\; \approx \; -->0.86\mathrm{\%<!--\; \%\; -->}$

$13:3$

$12:4=3:1$

The odds for $E$ are $4:1$ and the odds against $E$ are $1:4$.

$P(E)=\frac{1}{16}$

$P(E)=\frac{2.5}{3.5}\approx <!--\; \approx \; -->0.714$

Mutually exclusive

Not mutually exclusive

Mutually exclusive

$\frac{4}{10}+\frac{2}{10}=\frac{3}{5}$

Not appropriate; the events are not mutually exclusive.

$\frac{3}{10}+\frac{2}{10}=\frac{1}{2}$

$\frac{1}{2}$

$\frac{1}{2}$

$\frac{2}{3}$

$\frac{1}{3}$

$1$

$\frac{2}{3}$

$\frac{3}{28}$

$\frac{3}{56}$

$\frac{3}{28}$

Not binomial (more than two outcomes)

Not binomial (not independent)

Binomial

Not binomial (number of trials isn’t fixed)

0.146

0.190

0.060

0.2972

0.6615

0.0919

0.5207

0.3643

$\frac{10}{3}$

$\frac{3}{2}$

$\frac{25}{4}$ = $6.25

If you roll the special die many times, the mean of the numbers showing will be around 3.33.

If you repeat the coin-flipping experiment many times, the mean of the number of heads you get will be around 1.5.

If you play this game many times, the mean of your winnings will be around $10.

If the player bets on 7, the expected value is $-<!--\; -\; -->\mathrm{\$<!--\; \$\; -->}0.17$.

If the player bets on 12, the expected value is $-<!--\; -\; -->\mathrm{\$<!--\; \$\; -->}0.14$.

If the player bets on any craps, the expected value is $-<!--\; -\; -->\mathrm{\$<!--\; \$\; -->}0.11$.
The best bet for the player is any craps; the best bet for the casino is the bet on 7.

540

14

1,024

800

25,920

120

120

1,320

1,680

${}_{15}{P}_4=\mathrm{32,760}$

permutations

combinations

66

560

20

560

{0, 1, 2, 3, 4, 5, 6}

{8 nuggets with crinkle fries, 8 nuggets with curly fries, 8 nuggets with onion rings, 12 nuggets with crinkle fries, 12 nuggets with curly fries, 12 nuggets with onion rings}

{history with Anderson, history with Burr, English with Carter, sociology with Johnson, sociology with Kirk, sociology with Lambert}

$\frac{1}{4}$

$\frac{1}{2}$

$\frac{1}{2}$

0

theoretically

subjectively

empirically

$\text{number of heads}>20$

89.9%

$\frac{1\times <!--\; \times \; -->1\times <!--\; \times \; -->2}{{}_{10}{P}_3}\approx <!--\; \approx \; -->0.278\mathrm{\%<!--\; \%\; -->}$

$\frac{1\times <!--\; \times \; -->1\times <!--\; \times \; -->2\times <!--\; \times \; -->2}{{}_{10}{P}_4}\approx <!--\; \approx \; -->0.079\mathrm{\%<!--\; \%\; -->}$

$\frac{2\times <!--\; \times \; -->2\times <!--\; \times \; -->1\times <!--\; \times \; -->2}{{}_{10}{P}_4}\approx <!--\; \approx \; -->0.16\mathrm{\%<!--\; \%\; -->}$

$\frac{1\times <!--\; \times \; -->1\times <!--\; \times \; -->2}{{}_{10}{C}_3}\approx <!--\; \approx \; -->1.67\mathrm{\%<!--\; \%\; -->}$

$\frac{1\times <!--\; \times \; -->1\times <!--\; \times \; -->2\times <!--\; \times \; -->2}{{}_{10}{C}_4}\approx <!--\; \approx \; -->1.9\mathrm{\%<!--\; \%\; -->}$

$\frac{2\times <!--\; \times \; -->1\times <!--\; \times \; -->2}{{}_{10}{C}_4}\approx <!--\; \approx \; -->1.9\mathrm{\%<!--\; \%\; -->}$

$1:2$

$5:1$

$1:1$

$3:5$

$11:2$

$\frac{4}{13}$

$\frac{5}{12}$

$\frac{3}{5}$

$\frac{3}{5}$

$\frac{3}{5}$

$\frac{2}{5}$

$\frac{2}{5}$

$\frac{3}{10}$

$\frac{1}{4}$

$\frac{1}{9}$

$\frac{1}{6}$

$\frac{2}{15}$

$\frac{4}{45}$

$\frac{8}{45}$

No (more than two outcomes)

Yes

No (number of trials is not fixed)

0.9453

0.1366

0.8230

4.1

If you roll this die many times, the mean of the numbers rolled will be around 4.1.

$0.417

If you play this game many times, the mean of the amount won/lost each time will be about 42 cents.

Yes; the expected value is positive.