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Statistics

David Freedman, Robert Pisani, Roger Purves

Chapter 16

The Law of Averages - all with Video Answers

Educators

Chapter Questions

02:28

Problem 1

A box contains $10,000$ tickets: $4,000[0]$ 's and $6,000[11$ 's. And $10,000$ draws will be made at random with replacement from this box. Which of the following best describes the situation, and why?
(i) The number of 1 's will be $6,000$ exactly.
(ii) The number of $1^{\text { 's }}$ is very likely to equal $6,000,$ but there is also some small chance that it will not be equal to $6,000 .$
(iii) The number of $1^{\prime}$ 's is likely to be different from $6,000,$ but the difference is likely to be small compared to $10,000 .$

Bryan Luo
Bryan Luo
Numerade Educator
02:12

Problem 2

Repeat exercise 1 for $10,000$ draws made at random without replacement from the box.

Heena Haldankar
Heena Haldankar
Numerade Educator
View

Problem 3

A gambler loses ten times running at roulette. He decides to continue playing because he is due for a win, by the law of averages. A bystander advises him to quit, on the grounds that his luck is cold. Who is right? Or are both of them wrong?

Nick Johnson
Nick Johnson
Numerade Educator
06:31

Problem 4

(a) A die will be rolled some number of times, and you win $\$ 1$ if it shows an ace (.) more than 20$\%$ of the time. Which is better: 60 rolls, or 600 rolls? Explain.
(b) As in (a), but you win the dollar if the percentage of aces is more than 15$\% .$
(c) As in (a), but you win the dollar if the percentage of aces is between 15 $\%$ and 20$\% .$
(d) As in (a), but you win the dollari the percentage of aces is exactly 16$\frac{2}{3} \%$

Heena Haldankar
Heena Haldankar
Numerade Educator
01:31

Problem 5

True or false: if a coin is tossed 100 times, it is not likely that the number of heads will be exactly $50,$ but it is likely that the percentage of heads will be exactly 50$\% .$ Explain.

Bryan Luo
Bryan Luo
Numerade Educator
02:46

Problem 6

According to genetic theory, there is very close to an even chance that both children in a two-child family will be of the same sex. Here are two possibilities.
(i) 15 couples have two children each. In 10 or more of these families, it will turn out that both children are of the same sex.
(ii) 30 couples have two children each. In 20 or more of these families, it will turn out that both children are of the same sex.
Which possibility is more likely, and why?

Heena Haldankar
Heena Haldankar
Numerade Educator
02:17

Problem 7

A quiz has 25 multiple choice questions. Each question has 5 possible answers, one of which is correct. A correct answer is worth 4 points 4 points, but a point is taken off for each incorrect answer. A student answers all the questions by guessing at random. The score will be like the sum of box of tickets. Explain your answers.

Bryan Luo
Bryan Luo
Numerade Educator
02:29

Problem 8

A gambler will play roulette 50 times, betting a dollar on four joining numbers each time (like $23,24,26,27$ in figure $3,$ p. $282 ) .$ If one of these four numbers comes up, she gets the dollar back, together with winnings of $\$ 8 .$ If any other number comes up, she loses the dollar. So this bet pays 8 to $1,$ and there are 4 chances in 38 of winning. Her net gain in 50 plays is like the sum of $\longrightarrow$ draws from the box $\longrightarrow$ Fill in the blanks; explain.

Heena Haldankar
Heena Haldankar
Numerade Educator
01:41

Problem 9

A box contains red and blue marbles; there are more red marbles than blue ones. Marbles are drawn one at a time from the box, at random with replacement. You win a dollar if a red marble is drawn more often than a blue one. 6 There are two choices:
(A) 100 draws are made from the box.
(B) 200 draws are made from the box.
Choose one of the four options below; explain your answer.
(i) A gives a better chance of winning.
(ii) $B$ gives a better chance of winning.
(iii) $A$ and $B$ give the same chance of winning.
(iv) Can't tell without more information.

Bryan Luo
Bryan Luo
Numerade Educator
02:54

Problem 10

Two hundred draws will be made at random with replacement from the box
(a) If the sum of the 200 numbers drawn is $30,$ what is their average?
(b) If the sum of the 200 numbers drawn is $-20,$ what is their average?
(c) In general, how can you frawn is $-20,$ what is their average? told their sum?
(d) There are two alternatives:
(i) winning $\$ 1$ if the sum of the 200 numbers drawn is between $-5$ and $+5 .$
(ii) winning $\$ 1$ if the average of the 200 numbers drawn is between $-0.025$ and $+0.025 .$
Which is better, or are they the same? Explain.

Heena Haldankar
Heena Haldankar
Numerade Educator