A box contains 5 green marbles, 8 blue marbles, and 8 red marbles. Three marbles are...

A box contains 5 green marbles, 8 blue marbles, and 8 red marbles.

Three marbles are selected at random from the box, one at a time, without replacement.

Find the probability that the first two marbles selected are not red, and the last marble is red.

Solutions

Expert Solution

box contains 5 green marbles, 8 blue marbles, and 8 red marbles

Total = 8 + 8 + 5 = 21

P[ first two marbles selected are not red, and the last marble is red ] = P[ first two marbles selected are blue, and the last marble is red ] + P[ first two marbles selected are green, and the last marble is red ] + 2*P[ first two marbles selected are one blue and one green and the last marble is red ]

( 2 is multiplied because there are two possibilities (green, blue) and (blue,green )

P[ first two marbles selected are blue, and the last marble is red ] = 8C2*8C1/21C3

P[ first two marbles selected are blue, and the last marble is red ] = (8*7*8/2)/21C3

P[ first two marbles selected are blue, and the last marble is red ] = 224/21C3

P[ first two marbles selected are green, and the last marble is red ] = 5C2*8C1/21C3

P[ first two marbles selected are green, and the last marble is red ] = (5*4*8/2)/21C3

P[ first two marbles selected are green, and the last marble is red ] = 80/21C3

P[ first two marbles selected are one blue and one green and the last marble is red ] = 5C1*8C1*8C1/21C3

P[ first two marbles selected are one blue and one green and the last marble is red ] = 8*8*5/21C3

P[ first two marbles selected are one blue and one green and the last marble is red ] = 320/21C3

P[ first two marbles selected are not red, and the last marble is red ] = ( 224 + 80 + 2*320 )/ 21C3

P[ first two marbles selected are not red, and the last marble is red ] = 944 / ( 21*20*19/3*2*1)

P[ first two marbles selected are not red, and the last marble is red ] = 944/1330

P[ first two marbles selected are not red, and the last marble is red ] = 0.7098

orchestra answered 1 year ago

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