Question

If 3P(A) = P(B) = `3/5` and P(A | B) = `2/3`, then P(A∪B) is ______.

Options

• `3/5`

• `1/5`

• `2/15`

• `2/5`

Answer 1

If 3P(A) = P(B) = `3/5` and P(A | B) = `2/3`, then P(A∪B) is `bbunderline(2/5)`.

Explanation:

We use the addition rule of probability:

P(A∪B) = P(A) + P(B) − P(A∩B)

Step 1: Find P(A) and P(B)

From the given equation 3P(A) = P(B) = `3/5`

P(B) = `3/5`

3P(A) = `3/5`

⇒ P(A) = `3/(5 xx 3)`

= `1/5`

Step 2: Find P(A∩B)

We use the formula for conditional probability:

P(A | B) = `(P(A∩B))/(P(B))`

`2/3 = (P(A∩B))/(3/5)`

P(A∩B) = `2/3 xx 3/5`

= `2/5`

Step 3: Calculate P(A∪B)

Substitute the values into the addition rule:

P(A∪B) = P(A) + P(B) − P(A∩B)

= `1/5 + 3/5 - 2/5`

= `(1 + 3 − 2)/5`

= `2/5`

P(A∪B) is `2/5`