Suppose that an experiment has five possible outcomes, which are denoted {1, 2,

Question

Suppose that an experiment has five possible outcomes, which are denoted {1, 2, 3, 4. 5}. Let A be the event {1, 2, 3} and let B be the event {3, 4, 5}. (Notice that we did not say that the five outcomes are equally likely: the probability distributions could be anything.) For each of the following relations, tell whether it could possibly hold. If it could, give a numerical example using a probability distribution of your own choice: if it could not, explain why not (what rule is violated). a. P(A) = P(B) b. P(A) = 2P(B) c. P(A) = 1 - P(B) d. P(A) + P(B) > 1 e. P(A) - P(B) 1

Explanation / Answer

A) P(A) = P(B) Can Hold:

Example if set is uniformly distributed. P(A) = 0.6 = P(B)

B) P(A) = 2P(B) Can hold:

P(1)=.035 P(2)=0.3 P(3)=0.05 P(4)=0.15 P(5)=0.15

since 0.7 = 0.7

C) P(A) = 1 - P(B)

P(A) = P(1) + P(2) +P(3),

P(B') = P(1) + P(2) This holds if P(3) = 0

D) P(A) + P(B) > 1 Can hold if uniformly distributed. 0.6+0.6 = 1.2

E) P(A) - P(B) < 0 Can hold:

P(1) = 0.1, P(2) = .1, P(3) = .2, P(4) = .3, P(5) = .3

P(A)= 0.4, P(B)=0.8

P(A) - P(B) = -0.4

F) P(A) - P(B) > 1

Can not hold. P(A) & P(B) are both between 1 and 0.

Using extreme points 1-1 = 0, 0-1 = -1, 1-0 = 1.

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