Question

Box X contains 3 red, 2 white marbles; box Y contqains red, white marbles A box os selected at random; a marble is drawn and put into the other box; then a marble is drawn from the second box Find the probability that both marbles drawn are of the same color

Answer 1

Box X : 3 red , 2 white

Box Y : 1 red, 1 white

case 1 : p = 1/2 , box X selected first

case 1.1 :

P(red from box X) = 3/5

then red transfered to box y now box y : 2 red , 1 white

P(red from box y) = 2/3

P(both red | case 1) = 3/5 * 2/3 = 2/5

case 1.2 :

P(white from box X) = 2/5

then white transfered to box y now box y : 1 red , 2 white

P(white from box y) = 2/3

P(both white | case 1) = 2/5 * 2/3 = 4/15

case 2 : p = 1/2 , box Y selected first

case 2.1 :

P(red from box Y) = 1/2

then red transfered to box x, now box x : 4 red , 2 white

P(red from box x) = 4/6 = 2/3

P(both red | case 2) = 1/2 * 2/3 = 1/3

case 2.2 :

P(white from box Y) = 1/2

then white transfered to box x, now box x : 3 red , 3 white

P(white from box x) = 3/6 = 1/2

P(both white | case 2) = 1/2 * 1/2 = 1/4

P(both same color) = P(case 1)*(P(both red | case 1) + P(both white | case 1)) + P(case 2)*((P(both red | case 2) + P(both white | case 2)) )

= 1/2 * (2/5 + 4/15) + 1/2 * (1/3 + 1/4)

= 0.625

P(both same color) = 0.625

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