Question

IA Box 1 contains 5 red and 4 green balls and box 2 contains 4 red and 6 green balls. Three balls are randomly drawn from box 1 and placed in box 2. Then a ball is taken from box 2. If the ball taken from box 2 is found red, find the probability that 2 red and 1 green balls are transferred from box 1 to box 2? IB. Two coins Ci and C2 have a probability of falling heads p, and p2, respectively. You win a bet if in 3 tosses you get at least two heads in succession. You toss the 3 coins alternately starting with either coin. If p,>P2, what coin would you select to start the game?

Answer 1

Part 1A
Box 1 has 5 red and 4 green balls. Total 9 bals.
We will transfer 3 balss out of these 9, to box 2 and need to find the probability that out of these 3 balls, 2 are red and 1 is green. The whole situation of transferring these balls to Box and then picking one red ball back is given only to confuse.

We can pic these three balss in the following possible combinations

RGR
GRR
RRG

During the first draw, the total number of balls in the Box 1 is 9, so the denominator for first draw will be 9.
During the second draw, the total number of balls in the Box 1 is 8, so the denominator for first draw will be 8.
During the third draw, the total number of balls in the Box 1 is 7, so the denominator for first draw will be 7.  

Probability of getting 2 red and 1 green bals = P (RGR) + P (GRR) + P (RRG)

P (RGR) = 5/9 (5 red balls/ total in box) * 4/8 (4 green balls/ total in box) * 4/7 (4 red bals remaining in box/ total in box)
P (RGR) = (5* 4* 4)/ (9*8*7)
P (RGR) = 80/504

Similarly

P (GRR) = 4/9 * 5/8 * 4/7
= 80/504

P (RRG) = 5/9 * 4/8 * 4/7
= 80/504

Probability of getting 2 red and 1 green bals = P (RGR) + P (GRR) + P (RRG)

= 80/ 504 + 80/504 + 80/504
= 240/ 504
= 0.476

part 1B
We will win if we get 2 heads in succession.
This means that the coin with greater probabolity of getting a heads is most likely to give a victory.

Since P1 (probability of getting heads with coin C1) > P2 (probability of grtting heads with coin C2),
We will choose C1 to begin with.