Lactose intolerance Lactose intolerance affects about $20 \%$ of non-Hispanic White Americans; $50 \%$ of Hispanic Americans; and $75 \%$ of African, Asian, and Native Americans (Source: Jean Carper, "Eat Smart," USA Weekend). Seventy-six percent of U.S. residents are non-Hispanic Whites; $9 \%$ of them are Hispanic; and $15 \%$ are African, Asian, or Native American. If a person is selected from this group of people, what is the probability that the person will have lactose intolerance?

An urn contains $4$ red and $6$ white balls. One ball is drawn, it is not replaced, and a second ball is drawn. What is the probability that one ball is white and one is red?

A retailer purchases $100$ of a new brand of $\mathrm{DVD}$ player, of which $2$ are defective. The purchase agreement says that if he tests $5$ chosen at random and finds $1$ or more defective, he receives all the $\mathrm{DVD}$ players free of charge. What is the probability that he will not have to pay for the $\mathrm{DVD}$ players?

Compute ${ }_{80} C_{76}$.

Use (a) a probability tree and (b) Bayes' formula to find the probabilities.

In the previous problem, what is the probability that urn $\mathrm{III}$ was selected if the coin selected was silver?

Computer chips come from two suppliers, with $80 \%$ coming from supplier $1$ and $20 \%$ coming from supplier $2$. Six percent of the chips from supplier $1$ are defective, and $8 \%$ of the chips from supplier $2$ are defective. If a chip is chosen at random, what is the probability that it is defective?

Suppose that 6 transistors are drawn at random from a box containing 18 good transistors and $2$ defective transistors. What is the probability that (a) all $6$ of the transistors are good? (b) exactly $4$ of the transistors are good? (c) exactly $2$ of the transistors are good?

If a green die and a red die are rolled, find

(a) $\operatorname{Pr}(2 \leq S \leq 6)$, where $S$ is the sum rolled on the two dice.

(b) $\operatorname{Pr}(4 \leq S)$, where $S$ is the sum rolled on the two dice.

A cube has $2$ faces painted red, $2$ painted white, and $2$ painted blue. What is the probability of getting a red face or a white face in one roll?

A bag contains $3$ red, $5$ black, and $8$ white balls. Three balls are drawn, without replacement. What is the probability that one of each color is drawn?

Use the matrix and initial probability vector in Problem $12$ and find the resulting ninth probability vector.

(a) One card is drawn at random from a deck of $52$ cards. The first card is replaced, and a second card is drawn. Find the probability that both are hearts.

(b) Answer the question in part (a) if the first card is not replaced before the second card is drawn.

If a pair of dice, one green and one red, is rolled, what is

(a) $\operatorname{Pr}(4 \leq S \leq 7)$, where $S$ is the sum rolled?

(b) $\operatorname{Pr}(8 \leq S \leq 12)$, where $S$ is the sum rolled?

In a game where only one player can win, the probability that Jack will win is $1 / 5$ and the probability that Bill will win is $1 / 4$. Find the probability that one of them will win.