Probability Question experiment

[USN2ENG]

Homework Statement

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) < 0
f. P(A) - P(B) > 1

Homework Equations

The Attempt at a Solution

I am honestly not sure If I am doing these right, so please correct me if I am in the wrong direction.

a. P(A) = P(B)
Can Hold: Example if set is uniformly distributed.
P(A) = .6 = P(B)

b. P(A) = 2P(B)
Can hold:
P(1)=.35
P(2)=.3
P(3)=.05
P(4)=.15
P(5)=.15

.7 = .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.
.6+.6 = 1.2

e. P(A) - P(B) < 0
Can hold: P(1) = .1, P(2) = .1, P(3) = .2, P(4) = .3, P(5) = .3
P(A)= .4, P(B)=.8
P(A) - P(B) = -.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.

 

[mfb]

They are all right.