Question

Use a probability tree to find the probabilities of the indicated outcomes.

Three balls are drawn, without replacement, from a bag containing 7 red balls and 6 white balls. (Enter your probabilities as fractions.)

(a) Find the probability that three white balls are drawn.

(b) Find the probability that two white balls and one red ball are drawn.

(c) Find the probability that the third ball drawn is red.

Answer 1

Total Balls = 7 Red + 6 White = 13

(a) All three are white:

To pick the 1st white ball from the 6 white and overall 13 balls = 6 / 13

To pick the 2nd white ball from the remaining 5 white and overall 12 balls (as one white is picked) = 5/12

To pick the 3rd white ball from the remaining 4 white and overall 11 balls (as 2 whites are picked) = 4/11

Therefore the required probability = (6/13) * (5/12) * (4/11) = 10 / 143

(b) 2 white and 1 Red:

To pick the 1st white ball from the 6 white and overall 13 balls = 6 / 13

To pick the 2nd white ball from the remaining 5 white and overall 12 balls (as one white is picked) = 5/12

To pick the 3rd Ball which is Red from 7 Red and overall 11 balls (as 2 whites are picked) = 7/11

Therefore the required probability = (6/13) * (5/12) * (7/11) = 35 / 286

(c) The Last ball is red:

The possible Picks are RRR or WRR or WWR or RWR

(i) For Red, Red, Red, the required probability = (7/13) * (6/12) * (5/11) = 210 / 1716

(ii) For White, Red, Red, the required probability = (6/13) * (7/12) * (6/11) = 252 / 1716

(iii) For White, White, Red, the required probability = (6/13) * (5/12) * (7/11) = 210 / 1716

(iv) For Red, White, Red, the required probability = (7/13) * (6/12) * (6/11) = 252 / 1716

Therefore the probability that the last ball is red = 210/1716 + 252/1716 + 210/1716 + 252/1716 = 924 / 1716 = 7 / 13