91Ó°ÊÓ

a. How many bit strings consist of from one through four digits? (Strings of different lengths are considered distinct. Thus 10 and 0010 are distinct strings.) b. How many bit strings consist of from five through eight digits?

In 1-4, use the fact that in baseball's World Series, the first team to win four games wins the series. Suppose team \(A\) wins the first three games. How many ways can the series be completed? (Draw a tree.)

a. According to Theorem \(6.5 .1\), how many 5 -combinations with repetition allowed can be chosen from a set of three elements? b. List all of the 5 -combinations that can be chosen with repetition allowed from \(\\{1,2,3\\}\).

In 1-4, use the fact that in baseball's World Series, the first team to win four games wins the series. Suppose team \(A\) wins the first two games. How many ways can the series be completed? (Draw a tree.)

a. How many strings of hexadecimal digits consist of from one through three digits? (Recall that hexadecimal numbers are constructed using the 16 digits \(0,1,2,3,4,5,6\), \(7,8,9, \mathrm{~A}, \mathrm{~B}, \mathrm{C}, \mathrm{D}, \mathrm{E}, \mathrm{F})\) b. How many strings of hexadecimal digits consist of from two through five digits?

a. According to Theorem \(6.5 .1\), how many multisets of size four can be chosen from a set of three elements? b. List all of the multisets of size four that can be chosen from the set \(\\{x, y, z\\}\).

In 1-4, use the fact that in baseball's World Series, the first team to win four games wins the series. How many ways can a World Series be played if team \(A\) wins four games in a row?

a. How many integers from 1 through 999 do not have any repeated digits? b. What is the probability that an integer chosen at random from 1 through 999 has at least one repeated digit?

a. Prove that if \(A\) and \(B\) are any events in a sample space \(S\), with \(P(B) \neq 0\), then \(P\left(A^{c} \mid B\right)=1-P(A \mid B)\). b. Explain how this result justifies the following statements: (1) If the probability of a false negative on a test for a condition is \(4 \%\), then there is a \(96 \%\) probability that a person who does not have the condition will have a negative test result. (2) If the probability of a false positive on a test for a condition is \(1 \%\), then there is a 995 probability that a person who does have the condition will test positive for it.

A bakery produces six different kinds of pastry. a. How many different selections of twenty pastries are there? b. Assuming that eclairs are one kind of pastry produced, how many different selections of twenty pastries are there if at least three must be eclairs? c. If a selection of twenty pastries is chosen randomly, what is the probability that at least three are eclairs? d. If a selection of twenty pastries is chosen randomly, what is the probability that exactly three are eclairs?