Two patients share a hospital room for two days. Suppose that, on any given day, a person independently picks up an airborne infection with probability 1/4. An individual who is infected on the first day will certainly pass it to the other patient on the second day. Once contracted, the infection stays for at least two days. What is the probability that fewer than two patients have the infection by the end of the second day? 

Suppose 1/10 of the population has a disease. If a person has the disease, then a test detects it with probability 8/10. If a person does not have the disease, then the test incorrectly shows the presence of the disease with probability 2/10. What is the probability that the person tested has the disease if the test indicates the presence of the disease?

There are 3 red and 5 black balls in an urn. You draw two balls in succession without replacing the first ball.
The probability that the second ball is red equals

An insurance policy-holder can submit up to 5 claims. The probability that the policyholder submits exactly n claims is p_n, for n = 0, 1, 2, 3, 4, 5. It is known that
(a)  The difference between p_n and p_{n + 1} is constant for n = 0, 1, 2, 3, 4, and 

(b)  40% of the policyholders submit 0 or 1 claim.
What is the probability that a policy-holder submits 4 or 5 claims?

A tour operator has a bus that can accommodate 20 tourists. The operator knows that tourists may not show up, so he sells 21 tickets. The probability that an individual tourist will not show up is 0.02, independent of all other tourists. Each ticket costs 50, and is non-refundable if a tourist fails to show up. If a tourist shows up and a seat is not available, the tour operator has to pay 100 to that tourist. The expected revenue of the tour operator is 

In a roll of two fair dice, X is the number on first die and Y is the number on second die. Which of the following statements is True

A family has two children, what is the probability that both are girls given that at least one child is girl?

A student is answering a multiple‐choice examination. Suppose a question has m possible answers. The student knows the correct answer with probability p. If the student knows the correct answer, then she picks that answer; otherwise, she picks randomly from the choices with probability 1/m each. Given that the student picked the correct answer, the probability that she knew the correct  answer is

Suppose two fair dice are tossed simultaneously. What is the probability that the total number of spots on the upper faces of the two dice is not divisible by 2, 3, or 5?

A coin toss has possible outcomes H and T with probabilities 3/4 and 1/4 respectively. A gambler observes a sequence of tosses of this coin until H occurs. Let the first H occur on the n^th toss. If n is odd, then the gambler’s prize is −2ⁿ , and if n is even, then the gambler’s prize is 2ⁿ . What is the expected value of the gambler’s prize?