Suppose you are answering a multiple-choice problem on a exam, and have to choose one ofn options (exactly one of which is correct). Let K = the event that you know the answer,and R = the event that you get the problem right. Suppose that if you know the right answer, you will definitely getthe problem right, but if you do not know, then you will guess completely randomly. LetP (K) = some probability p.(a)  Find P (K|R) (in terms of p and n).(b)  Prove that P (K|R) ≥ p, and give a more intuitive explanation for why thisis true.(c)  When (if ever) does P (K|R) equal p?

If we know P(A and B) = P(A) x P(B | A), we can say event A and B are independent

= Lisa is flying from Boston to Denver with a connection in Chicago. The probability her first flight leaves on time is 0.25. If the flight is on time, the probability that her luggage will make the connecting flight is 0.9, but if the flight is delayed, the probability that the luggage will make it is only 0.65. Suppose you pick her up at the Denver airport and her luggage is not there. What is the probability that Lisa's first flight was delayed? P(delayed) = (Do not round until the final answer. Then round to three decimal places as needed.)

Suppose that A and B are two events for which P(A)=0.31, P(B)=0.68, and P(AandB)=0.31 Find P(A|B). P(A|B)=

Suppose that 20,000 married adults in a country were randomly surveyed as to the number of children they have. The results are compiled and are used as theoretical probabilities. Let X = the number of children married people have. P(x) ХP(x) 0.15 1 0.25 0.35 4 0.10 0.05 6 (or more) 0.05 (a) Find the probability that a married adult has three children. (Enter your answer to two decimal places.) (b) In words, what does the expected value in this example represent? O the average number of children married adults in the country have the average number of children adults in the country have O the number of children married adults in the country have O the number of children adults in the country have (c) Find the expected value. (Enter your answer to two decimal place.) children (d) Is it more likely that a married adult will have two to three children or four to six children? How do you know? O it is more likely to have two to three children, with p = 0.35 O it is more likely to have four…

O Whats inside of blo. S Watch Cra zy Rich A Car inspection: Of all the registered automobiles in a city, 6% fail the emissions test. Nine automobiles are selected at random to undergo an emissions test. Round the answers to at least four decimal places. Part 1 of 4 (a) Find the probability that exactly four of them fail the test. The probability that exactly four of them fail the test is Part 2 of 4 (b) Find the probability that fewer than four of them fail the test. The probability that fewer than four of them fail the test is Part 3 of 4 (c) Find the probability that more than three of them fail the test., The probability that more than three of them fail the test is

Let A and B be two events such that the probability of B is 0.5 and the probability of both is 0.2. Find the probability of A given that B occurs.

Suppose that A and B are two events for which P(A)=0.22, P(B)=0.82, and P(AandB)=0.36 Find P(A|B).