4 A and B are playing a short game of ping pong where they e

4) A and B are playing a short game of ping pong where they each serve twice. If after these four points one of them is ahead by at least two points the game ends, otherwise they go into a second phase. Suppose that A wins 70% of the time when they serve and 40% of the time when B serves. Let?s look at the first phase.

a) \\Find the probability that A scores 0, 1, or 2 points when they serve and when B serves.

b) Find the probability that A scores a total of 3 or 4 points (so wins in the first phase).

c) Find the probability that A scores 2 points in total (so there is a tie in the first phase).

Now lets look at cases where the game moves on to the second phase. In this phase there are multiple rounds: in each round each player serves once. They win if they win both points; otherwise it goes to another round. Play continues until someone wins.

d) Find the probability that A wins if it goes to the second phase.

e) Find the probability that A wins (in either the first or second phase).

Solution

a) There are 2 marbles numbered 1 and 2 marbles numbered 2

P(The number is odd)

After the first draw we put the marble along with the same number hence result will always be even.

For example first marble is 1 then placed along with 1, sum is 2.

first is 2, placed along with 2, sum is 4.

Hence both even.

II marble: if it is 1, then sum will be either 3 or 5.

Then sum after II draw will be either 2+1+2 =5 or 2+2+3=7

Thus both cases it is odd.

If second marble is 2, then sum after II draw will be either 1+1+2+3 =7 or 2+2+2+4 = 10

a) Prob for sum odd = 1/4 = 0.25

b) Prob (first number was 2 | sum=odd) = 0.5(0.25) =0.125