SOLUTION: A store purchases baseball hats from three different manufacturers. In manufacturer A's box, there are 12 blue hats, 6 red hats, and 6 green hats. In manufacturer B's box, there ar

Algebra ->  Probability-and-statistics -> SOLUTION: A store purchases baseball hats from three different manufacturers. In manufacturer A's box, there are 12 blue hats, 6 red hats, and 6 green hats. In manufacturer B's box, there ar      Log On


   



Question 1200461: A store purchases baseball hats from three different manufacturers. In manufacturer A's box, there are 12 blue hats, 6 red hats, and 6 green hats. In manufacturer B's box, there are 10 blue hats, 10 red hats, and 4 green hats. In manufacturer C 's box, there are 8 blue hats, 8 red hats, and 8 green hats. A box is selected at random, and a hat is selected at random from that box. If the hat is red, find the probability that it came from manufacturer A's box.

Answer by ikleyn(52777) About Me  (Show Source):
You can put this solution on YOUR website!
.
A store purchases baseball hats from three different manufacturers.
In manufacturer A's box, there are 12 blue hats, 6 red hats, and 6 green hats.
In manufacturer B's box, there are 10 blue hats, 10 red hats, and 4 green hats.
In manufacturer C 's box, there are 8 blue hats, 8 red hats, and 8 green hats.
A box is selected at random, and a hat is selected at random from that box.
If the hat is red, find the probability that it came from manufacturer A's box.
~~~~~~~~~~~~~~~~


            This problem is on conditional probability.


They want you find the probability that a red hat is from box A given that
it is red, P(red hat is from box A | the hat is red).


This probability is the ratio

    P(red hat is from box A | the hat is red) = P%28red_hat_is_from_box_A%29%2FP%28the_hat_is_red%29.  (*)


The numerator is

    P(red hat is from box A) = %281%2F3%29%2A%286%2F%2812%2B6%2B6%29%29 = %281%2F3%29%2A%286%2F24%29 = 1%2F12.


The denominator is 

    P(the_hat_is_red) =  = 

                      = %281%2F3%29%2A%286%2F24+%2B+10%2F24+%2B+8%2F24%29 = %281%2F3%29%2A%2824%2F24%29 = 1%2F3.


Therefore, the final probability (*) is

    P(red hat is from box A | the hat is red) = %28%281%2F12%29%29%2F%28%281%2F3%29%29 = 3%2F12 = 1%2F4.    ANSWER

Solved.