Question

There are two identical urns containing respectively 6 black and 4 red balls, 2 black and 2 red balls. An urn is chosen at random and a ball is drawn from it. find the probability that the ball is black

Answer 1

	Black balls	Red balls	Total
Urn I	6	4	10
Urn I	2	2	4
Total	8	6	14

Let A₁ be the event of selecting Urn – I and A₂ be the event of selecting Urn – II.

Let B be the event of selecting one black ball.

We have to find the total probability of event B.

That is P(B).

Clearly, A₁ and A₂ are mutually exclusive and exhaustive events.

Probability of selecting Urn – I

P(A₁) = `1/2`

Conditional Probability of B, given A₁ 

P(B/A₁) = `(""^6"C"_1)/(""^10"C"_1)`

= `6/10`

= `3/5`

Probability of selecting Urn – II

P(A₂) = `1/2`

Conditional Probability of B, given A₂ 

P(B/A₂) = `(""^2"C"_1)/(""^4"C"_1)`

=  `2/4`

= `1/2`

We know P(B) = P(B/A₁) . P(B/A₁) + P(A₂) . P(B/A₂) 

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

= `3/10 + 1/4`

= `(6 + 5)/20`

= `11/20`