Question

If P(B) = `3/5`, P(A|B) = `1/2` and P(A∪ B) = `4/5`, then P(A∪ B)′ + P( A′ ∪ B) = ______.

पर्याय

• `1/5`

• `4/5`

• `1/2`

• 1

Answer 1

If P(B) = `3/5`, P(A|B) = `1/2` and P(A∪ B) = `4/5`, then P(A∪ B)′ + P( A′ ∪ B) = 1.

Explanation:

Given that: P(B) = `3/5`, P(A|B) = `1/2` and P(A∪ B) = `4/5`

`"P"("A"/"B") = ("P"("A" ∩ "B"))/("P"("B"))`

⇒ `1/2 = ("P"("A" ∩ "B"))/(3/5)`

⇒ P(A ∩ B) = `3/10`

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

`4/5 = "P"("A") + 3/5 - 3/10`

∴ P(A) = `4/5 - 3/5 + 3/10`

= `1/5 + 3/10`

= `5/10`

= `1/2`

Now P(A ∪ B)' + P(A' ∪ B)

= 1 – P(A ∪ B) + 1 – P(A ∩ B')

= `2 - 4/5 - "P"("A")*"P"("B'")`

= `6/5 - 1/2 * (1 - 3/5)`

= `6/5 - 1/2 xx 2/5`

= `6/5 - 1/5`

= `5/5`

= 1