Question

There are two bags.the first bag contain five white balls and 3 black balls and the second bag contain three white and five black balls .two balls are drawn at random from the first bag and put it in the second bag without noticing their colours then two balls are drawn from the second bag .Find the probability that the balls are white and black.

Answer 1

probability of white :5/8
probability of black :3/8

Answer 2

probability that the balls are white and black. = 673/1260

Step-by-step explanation:

First Bag

White Balls = 5

Black Balls = 3

Total Balls = 8

Second Bag

White Balls = 3

Black Balls = 5

Total Balls = 8

Two balls are drawn at random from the first bag and put it in the second bag

Three cases:

1 . Both Balls Drawn from 1st Bag are White

2. Both Balls Drawn from 1st Bag are Black

3. Balls Drawn from 1st Bag are Black & white

Case1 :

Both Balls Drawn from 1st Bag are White

⁵C₂/⁸C₂ = 10/28 = 5/14

Then 2nd bag will have 5 White Balls  5 Black balls & Total 10 Balls

probability that the balls are white and black. = ⁵C₁ * ⁵C₁ / ¹⁰C₂ = 5*5/45 = 5/9

Probability case 1 = (5/14) * (5/9) = 25/126

Case2 :

Both Balls Drawn from 1st Bag are Black

³C₂/⁸C₂ = 3/28

Then 2nd bag will have 3 White Balls  7 Black balls & Total 10 Balls

probability that the balls are white and black. = ³C₁ * ⁷C₁ / ¹⁰C₂ = 21/45 = 7/15

Probability case 2 = (3/28) * (7/15) = 1/20

Case3 :

Balls Drawn from 1st Bag are Black & white

⁵C₁ * ³C₁/⁸C₂ = 5 * 3/28 = 15/28

Then 2nd bag will have 4 White Balls 6 Black balls & Total 10 Balls

probability that the balls are white and black. = ⁴C₁ * ⁶C₁ / ¹⁰C₂ = 24/45 = 8/15

Probability case 3 = (15/28) * (8/15) = 2/7

probability that the balls are white and black. = 25/126 + 1/20   + 2/7

= (250 + 63 + 360)/1260

= 673/1260

probability that the balls are white and black. = 673/1260

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