Game theory with alternating players

6 Tom and Ben play a game repeatedly. The probability that Tom wins any game is 0.3 . Each game is won by either Tom or Ben. Tom and Ben stop playing when one of them (to be called the champion) has won two games.

1. Find the probability that Ben becomes the champion after playing exactly 2 games.
2. Find the probability that Ben becomes the champion.
3. Given that Tom becomes the champion, find the probability that he won the 2nd game.

10 Jill and Kate are playing a game as a practice for a penalty shoot-out. They take alternate turns at kicking a football at a goal. The probability that Jill will score a goal with any kick is \(\frac { 1 } { 3 }\), independently of previous outcomes. The probability that Kate will score a goal with any kick is \(\frac { 1 } { 4 }\), independently of previous outcomes. Jill begins the game. If Jill is the first to score, then Kate is allowed one more kick. If Kate scores with this kick, then the game is a draw, but if she does not score then Jill wins the game. If Kate is the first to score, then she wins the game, and no further kicks are taken.

1. Show that the probability that Jill scores on her 5th kick is \(\frac { 1 } { 48 }\).
2. Show that the probability that Kate wins the game on her \(n\)th kick is \(\frac { 1 } { 3 \times 2 ^ { n } }\).
3. Find the probability that Jill wins the game.
4. Find the probability that the game is a draw.

1. Suzanne and Jon are playing a game.

They put 4 red counters and 1 blue counter in a bag.
Suzanne reaches into the bag and selects one of the counters at random. If the counter she selects is blue, she wins the game. Otherwise she puts it back in the bag and Jon selects one at random. If the counter he selects is blue, he wins the game. Otherwise he puts it back in the bag and they repeat this process until one of them selects the blue counter.

1. Find the probability that Suzanne selects the blue counter on her 4th selection.
2. Find the probability that the blue counter is first selected on or after Jon's third selection.
3. Find the mean and standard deviation of the number of selections made until the blue counter is selected.
4. Find the probability that Suzanne wins the game.

2. Mary and Jeff are archers and one morning they play the following game. They shoot an arrow at a target alternately, starting with Mary. The winner is the first to hit the target. You may assume that, with each shot, Mary has a probability 0.25 of hitting the target and Jeff has a probability \(p\) of hitting the target. Successive shots are independent.

1. Determine the probability that Jeff wins the game
   i) with his first shot,
   ii) with his second shot.
2. Show that the probability that Jeff wins the game is $$\frac { 3 p } { 1 + 3 p }$$
3. Find the range of values of \(p\) for which Mary is more likely to win the game than Jeff.

Sandra makes repeated, independent attempts to hit a target. On each attempt, the probability that she succeeds is 0.1.

1. Find the probability that
   1. the first time she succeeds is on her 5th attempt, [2]
   2. the first time she succeeds is after her 5th attempt, [2]
   3. the second time she succeeds is before her 4th attempt. [4]
   Jill also makes repeated attempts to hit the target. Each attempt of either Jill or Sandra is independent. Each time that Jill attempts to hit the target, the probability that she succeeds is 0.2. Sandra and Jill take turns attempting to hit the target, with Sandra going first.
2. Find the probability that the first person to hit the target is Sandra, on her
   1. 2nd attempt, [2]
   2. 10th attempt. [3]