SOLUTION: Three bags labelled P, Q and R contains red, blue and white balls respectively of equal sizes. The ratio of the balls in the bags are P:Q=2:1 and Q:R=4:5. All the balls are removed

Question 1120786: Three bags labelled P, Q and R contains red, blue and white balls respectively of equal sizes. The ratio of the balls in the bags are P:Q=2:1 and Q:R=4:5. All the balls are removed into a big bag and properly mixed together.
(a) Find the probability of pick a red ball.
(b) If two balls are picked at random one after the other with replacement, find the probability of picking:
(i) a white ball and a blue ball
(ii) a blue ball first and then a red ball.
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

When given the ratio of P:Q and the ratio of Q:R, you can get a ratio of all three by making the value of Q in both ratios the same:

P:Q = 2:1 and Q:R = 4:5 --> P:Q = 8:4 and Q:R = 4:5

Then P:Q:R = 8:4:5

Now, given that ratio,
let P = 8x
let Q = 4x
let R = 5x

Then after all the balls are mixed together, 8/17 of them are red (P), 4/17 are blue (Q), and 5/17 are white (R).

(a) The probability of picking a red ball is 8/17.

(b)(i) To pick a white and a blue with replacement, you can pick either a white first and then a blue, or a blue first and then white.
P(white, blue) = (5/17)(4/17) = 20/289
P(blue, white) = (4/17)(5,17) = 20/289
P(one blue and one white) = 40/289

(b)(ii) The probability of drawing a blue and then a red is (4/17)(8/17) = 32/289.

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