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