1. ## Conditional probability

An urn always contains 2 balls. Ball colors are red and blue. At each stage a ball is randomly
chosen and then replaced by a new ball, which with probability 0.8 is the same color, and with
probability 0.2 is the opposite color, as the ball it replaces. If initially both balls are red, find the
probability that the fifth ball selected is red.

2. ## Re: Conditional probability

General principle: If ball A has a Pa probability of being red and ball B has a Pb probability of being red, the probability of picking a red ball, Pr, is .5(Pa) + .5(Pb). If Pa=1 and Pb=1, the probability of picking a red ball, Pr, is 1.

First pick. Pr=1
After first replacement, Pa=.5x.8, Pb=.5x.8
(Pa has a 50% chance of being picked and an 80% chance of being red after replacement, and same for B.)

Second pick Pr = .5Pa +.5Pb = .5x.5x.8 +.5x.5x.8
After second replacement, Pa=.5x.5.x.8x.8 = Pb

Third pick Pr = .5Pa + .5Pb = .5x.5x.5x.8x.8 + .5x.5x.5x.8x.8
After third replacement, Pa = .5x.5x.5x.8x.8x.8 + .5x.5x.5x.8x.8x.8 = Pb

After fourth replacement, Pa = 2x(.5)^4x(.8)^4 = Pb

Fifth pick Pr = .5Pa + .5Pb = .5(Pa+Pb) = .5x4x(.5)^4x.8^4
Fifth pick Pr = 2x(.5)^4x(.8)^4 = .0512