If A, B, and C are mutually exclusive events with P(A) = 0.2, P(B) = 0.3, P(C) = 0.4, determine the following probabilities. a. P(A ∪ B ∪ C) b. P(A ∩ B ∩ C) c. P(A ∩ B) d. P[A ∪ B) ∩ C] e. P(A' ∩ B' ∩ C')

Let P(A) = 0.3 and P(B) = 0.6. a. Find P(A ∪ B) when A and B are independent. b. Find P(A | B) when A and B are mutually exclusive.

The sample space of a random experiment is {a, b, c, d, e} with probabilities 0.1, 0.1, 0.2, 0.4, and 0.2, respectively. Let A denote the event {a, b, c}, and let B denote the event {c, d, e}. Determine the following: a. P(A) b. P(B) c. P(A') d. P(A ∪ B) e. P(A ∩ B)

Let A and B be independent events with P(A) = 0.7 and P(B) = 0.2. Compute (a) P(A ∩ B), (b) P(A ∪ B), and (c) P(A' ∪ B').

E and F are mutually exclusive events. P(E) = 0.4; P(F) = 0.5. Find P(E|F).

Each of the possible five outcomes of a random experiment is equally likely. The sample space is {a, b, c, d, e}. Let A denote the event {a,b}, and let B denote the event {c, d, e}. Determine the following: a. P( A) b. P(B) c. P(A') d. P(A ∪ B) e. P(A ∩ B)

A and B are events for which P(A) = 0.3 + x, P(B) = 0.2 + x, and $P(A \cap B)=x$. Find x if A and B are mutually exclusive events.

Five men and 5 women are ranked according to their scores on an examination. Assume that no two scores are alike and all 10! possible rankings are equally likely. Let X denote the highest ranking achieved by a woman. (For instance, X=1 if the top-ranked person is female.) Find P{X = i}, i=1, 2, 3, ..., 8, 9, 10.

The events A and B are mutually exclusive. Suppose P(A) = .30 and P(B) = .20. What is the probability of either A or B occurring? What is the probability that neither A nor B will happen?

Given P(A) = 0.3 and P(B) = 0.4: If A and B are mutually exclusive events, compute P(A or B).

Let $A$ and $B$ be events with $P(A)=0.6, P(B)=0.3$, and $P(A \cap B)=0.2$. Find:

(a) $P(A \cup B)$; $\quad$ (b) $P(A \mid B)$; $\quad$ (c) $P(B \mid A)$.

Suppose that an insurance company classifies people into one of three classes — good risks, average risks, and bad risks. Their records indicate that the probabilities that good, average, and bad risk persons will be involved in an accident over a 1-year span are, respectively, .05, .15, and .30. If 20 percent of the population are “good risks,” 50 percent are “average risks,” and 30 percent are “bad risks,” what proportion of people have accidents in a fixed year? If policy holder A had no accidents in 1987, what is the probability that he or she is a good (average) risk?

A sample of two items is selected without replacement from a batch. Describe the (ordered) sample space for each of the following batches: a. The batch contains the items [a, b, c, d}. b. The batch contains the items {a, b, c, d, e, f, g}. c. The batch contains 4 defective items and 20 good items. d. The batch contains 1 defective item and 20 good items.

The independent events A and B are such that P(A) = 0.6 and

$$\mathrm { P } ( A \cup B ) = 0.72$$

. Find (a) P(B) (b) the probability that either A occurs or B occurs, but not both.