Mathematics

Problem

Let

S

be a sample space in which two events

A

and

B

satisfy

P(A)=0.5

and

P(B)=p

, where

0<p<1

. Suppose that

P(A\cup B)=0.7

. a. (i) Express

P(A\cap B)

in terms of

p

. b. (ii) Using the definition of independence, obtain the condition on

p

such that

A

and

B

are independent. c. (iii) For the value of

p

found in part (ii), prove that the complements

A'

and

B'

are also independent. d. (iv) For the same value of

p

, verify whether

A

and

B'

are independent. Show all steps clearly.

Intuitive explanation

The key is recognizing that the union formula $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ connects what you know (the individual probabilities and their union) to what you need to find. Since you're given three of the four quantities in this formula, you can solve for the missing piece—the overlap $P(A \cap B)$ —and then use it to determine $p$ . The constraint that $0 < p < 1$ will help you verify your answer makes sense.

Intuitive explanation

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