Q: A probability distribution is constructed from the number of TVs in 115 households. What is the mean number of TVs per household?

$$ \mu = \frac{0 \times 9 + 1 \times 25 + 2 \times 29 + 3 \times 19 + 4 \times 15 + 5 \times 8 + 6 \times 6 + 7 \times 4}{115} $$

Q: In a binomial experiment where 84% of students graduate and 10 students are sampled, what is the probability that exactly 3 students graduate?

$$ P(X=3) = \binom{10}{3} \times 0.84^3$ \times 0.16^7$ $$

Q: In the same binomial experiment, what is the mean number of students who graduate?

$$ \mu = n \times p = 10 \times 0.84 = 8.4 $$

Q: A raffle sells 1,000 tickets for $$1 each. The winner gets 500$$, everyone else gets nothing. What is the expected value for a person who buys one ticket?

$$ E = \frac{1}{1000} \times 500 + \frac{999}{1000} \times 0 - 1 = -0.5 $$

Q: In roulette, betting $$1 on a number has a probability of winning $$ \frac{1}{38} $$ and losing $$ \frac{37}{38} $$. The payoff is 35. $$What is the expected payoff?

$$ E = \frac{1}{38} \times 35 + \frac{37}{38} \times (-1) = -0.0526 $$

Q: To qualify for Calculus, a student must score in the top 15% on a placement test (mean 50, SD 4, normal distribution). What is the lowest possible qualifying score?

$$ x = \mu + z \sigma = 50 + (1.036) \times 4 = 54.14 $$

Question 1

A single card is drawn from a standard deck of 52 cards. What is the probability that the card is either Red or a Diamond?

Accepted Answer

$$ \frac{39}{52} $$

Question 2

At a used-book sale, there are 100 adult books (70 nonfiction) and 160 children’s books (100 fiction). What is the probability that a randomly selected book is a fiction book?

Accepted Answer

$$ \frac{130}{260} $$

Question 3

From the used-book sale scenario, what is the probability that a randomly selected book is both a children’s book and nonfiction?

Accepted Answer

$$ \frac{60}{260} $$

Question 4

From the used-book sale scenario, what is the probability that a randomly selected book is either a fiction book or an adult book?

Accepted Answer

$$ \frac{130 + 100 - 30}{260} = \frac{200}{260} $$

Question 5

A probability distribution is constructed from the number of TVs in 115 households. What is the mean number of TVs per household?

Accepted Answer

$$ \mu = \frac{0 	imes 9 + 1 	imes 25 + 2 	imes 29 + 3 	imes 19 + 4 	imes 15 + 5 	imes 8 + 6 	imes 6 + 7 	imes 4}{115} $$

Question 6

In a binomial experiment where 84% of students graduate and 10 students are sampled, what is the probability that exactly 3 students graduate?

Accepted Answer

$$ P(X=3) = \binom{10}{3} 	imes 0.84^3$ 	imes 0.16^7$ $$

Question 7

In the same binomial experiment, what is the mean number of students who graduate?

Accepted Answer

$$ \mu = n 	imes p = 10 	imes 0.84 = 8.4 $$

Question 8

A raffle sells 1,000 tickets for $$1 each. The winner gets 500$$, everyone else gets nothing. What is the expected value for a person who buys one ticket?

Accepted Answer

$$ E = \frac{1}{1000} 	imes 500 + \frac{999}{1000} 	imes 0 - 1 = -0.5 $$

Question 9

In roulette, betting $$1 on a number has a probability of winning $$ \frac{1}{38} $$ and losing $$ \frac{37}{38} $$. The payoff is 35. $$What is the expected payoff?

Accepted Answer

$$ E = \frac{1}{38} 	imes 35 + \frac{37}{38} 	imes (-1) = -0.0526 $$

Question 10

Replacement times for TV sets are normally distributed with mean 8.2 years and standard deviation 1.1 years. What is the probability that a randomly selected TV will be replaced within the first 5 years?

Accepted Answer

$$ P(X \u003c 5) = P\left(Z \u003c \frac{5 - 8.2}{1.1}\right) $$

Question 11

To qualify for Calculus, a student must score in the top 15% on a placement test (mean 50, SD 4, normal distribution). What is the lowest possible qualifying score?

Accepted Answer

$$ x = \mu + z \sigma = 50 + (1.036) 	imes 4 = 54.14 $$

Question 12

The weights of 1-year-old boys are normally distributed with mean 22.8 lbs and SD 2.15 lbs. What is the standard deviation of the sampling distribution of the sample mean for a sample of 10 boys?

Accepted Answer

$$ \sigma_{\bar{x}} = \frac{2.15}{\sqrt{10}} $$

Question 13

For the weights of 1-year-old boys (mean 22.8 lbs, SD 2.15 lbs, sample size 10), what is the probability that the sample mean is greater than 21 lbs?

Accepted Answer

$$ P\left(\bar{x} \u003e 21\right) = P\left(Z \u003e \frac{21 - 22.8}{2.15 / \sqrt{10}}\right) $$

Question 14

According to CreditCards.com, 29% of adults do not own a credit card. In a random sample of 500 adults, what is the mean and standard deviation of the sampling distribution of the sample proportion?

Accepted Answer

$$ \mu = p = 0.29 $$ and $$ \sigma = \sqrt{\frac{0.29 	imes 0.71}{500}} $$

Question 15

In the CreditCards.com scenario, what is the probability that between 25% and 30% of the sample of 500 adults do not own a credit card?

Accepted Answer

$$ P(0.25 \u003c \hat{p} \u003c 0.30) = P\left(\frac{0.25 - 0.29}{\sqrt{\frac{0.29 	imes 0.71}{500}}} \u003c Z \u003c \frac{0.30 - 0.29}{\sqrt{\frac{0.29 	imes 0.71}{500}}}ight) $$

Question 16

In the CreditCards.com scenario, would it be unusual for a random sample of 500 adults to result in 125 or fewer who do not own a credit card? Why?

Accepted Answer

Yes, because $$ \frac{125}{500} = 0.25  is $$more than 2 standard deviations below the mean.

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