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Black Jack (Posted on 2005-03-17) Difficulty: 2 of 5
Jack owns many black shirts and pants. However, Jack gets up late each day, and as a result, he just chooses his clothes at random.

He would like to have at least one black item on and he knows there is a .16 chance just his shirt will be black, there is a .27 chance just his pants will be black, and there is a chance less than both of these he won't have anything black on.

If the color of his shirt and pants are independent of each other, what is the chance both are black?

See The Solution Submitted by Gamer    
Rating: 3.0000 (2 votes)

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Solution Puzzle Solution Comment 8 of 8 |
(In reply to Answer by K Sengupta)

B = Probability that both his shirt and pants will be black.(say)

We know that:
Probability that  his shirt is black but the pants are white = 16/100, and:
Probability that  his shirt is white but the pants are black =  27/100

Then, we must have:

Probability that  his shirt is black
= Probability that both his shirt and pants will be black + Probability that  his shirt is black but the pants are white
=  B  + 16/100

Similarly, 

Probability that  his pants is black = B + 27/100

By the problem,

(B + 16/100) (B+27/100) = B
-> B^2 - 57*B/100 +  432/10000) = 0
-> 10000*B^2 - 5700*B + 432 = 0
-> 10000*B^2 - 4800*B - 900*B + 432 = 0
-> 100*B (100*B - 48) - 9(100*B - 48) = 0
-> (100*B - 48)(100*B - 9) = 0
-> B = 48/100, or 9/100

Now, the probability that at least one amongst the pants and the shirt is coloed black
= Prob(Black pants and white shirt) +  Prob(Black pants and black shirt) + prob(black shirt and white pants)
= 27/100 + B + 16/100
= B + 43/100

-> Prob.(none of the shirt and pants is black) = 1 -(B + 43/100)
= 57/100 - B

If B = 9/100, then Prob(black shirt and white pants) = 16/100 > 9/100.

This is a contradiction.

If B = 48/100, Prob(black shirt and white pants) = 16/100 < 48/100, and:
Prob(white shirt and black pants) = 27/100 < 48/100.

This is in conformity with the given conditions.

Hence B = 48/100 = 12/25

Consequently, the required chance that both the shirt and the pants are black is 12/25.


  Posted by K Sengupta on 2009-02-05 14:03:04
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