91Ó°ÊÓ

What is the expected value when a \(\$ 1\) lottery ticket is bought in which the purchaser wins exactly \(\$ 10\) million if the ticket contains the six winning numbers chosen from the set \(\\{1,2,3, \ldots, 50\\}\) and the purchaser wins nothing otherwise?

What is the probability that the sum of the numbers on two dice is even when they are rolled?

What is the probability that when a coin is flipped six times in a row, it lands heads up every time?

What is the expected sum of the numbers that appear when three fair dice are rolled?

Suppose that one person in \(10,000\) people has a rare genetic disease. There is an excellent test for the disease; 99.9\(\%\) of people with the disease test positive and only 0.02\(\%\) who do not have the disease test positive. a) What is the probability that someone who tests positive has the genetic disease? b) What is the probability that someone who tests negative does not have the disease?

Suppose that 8\(\%\) of the patients tested in a clinic are infected with HIV. Furthermore, suppose that when a blood test for HIV is given, 98\(\%\) of the patients infected with HIV test positive and that 3\(\%\) of the patients not infected with HIV test positive. What is the probability that a) a patient testing positive for HIV with this test is infected with it? b) a patient testing positive for HIV with this test is not infected with it? c) a patient testing negative for HIV with this test is infected with it? d) a patient testing negative for HIV with this test is not infected with it?

Suppose that we flip a fair coin until either it comes up tails twice or we have flipped it six times. What is the expected number of times we flip the coin?

What is the probability of these events when we randomly select a permutation of the 26 lowercase letters of the English alphabet? a) The first 13 letters of the permutation are in alphabetical order. b) \(a\) is the first letter of the permutation and \(z\) is the last letter. c) \(a\) and \(z\) are next to each other in the permutation. d) \(a\) and \(b\) are not next to each other in the permutation. e) \(a\) and \(z\) are separated by at least 23 letters in the permutation. f) \(z\) precedes both \(a\) and \(b\) in the permutation.

Suppose that we roll a fair die until a 6 comes up or we have rolled it 10 times. What is the expected number of times we roll the die?

Suppose that we roll a pair of fair dice until the sum of the numbers on the dice is seven. What is the expected number of times we roll the dice?