Curt agrees to play a game with his friend. He will risk half the chips he has in his pocket on the toss of a coin: heads he wins, tails he loses. Chips then go into his pocket or out of it. The game is repeated in eactly the same way. If after 4 tosses Curt has won twice and lost twice then he: a) won chips; b) lost chips; c) broke even; d) could have won or lost chips. (Hint try playing with 80 chips.)
d) could have won or lost chips.
Explanation:
Let's assume Curt starts with 80 chips, as suggested by the hint. We can represent the possible outcomes of each toss with the number of chips Curt wins or loses. Here's an example of each of the four possible sequences of wins and losses:
1. Win - Win - Lose - Lose
2. Win - Lose - Win - Lose
3. Lose - Win - Lose - Win
4. Lose - Lose - Win - Win
Now let's calculate how many chips Curt would have at the end of each sequence:
1. Win - Win - Lose - Lose: 80 + 40 = 120, 120 - 60 = 60, 60 - 30 = 30.
2. Win - Lose - Win - Lose: 80 + 40 = 120, 120 - 60 = 60, 60 + 30 = 90, 90 - 45 = 45.
3. Lose - Win - Lose - Win: 80 - 40 = 40, 40 + 20 = 60, 60 - 30 = 30, 30 + 15 = 45.
4. Lose - Lose - Win - Win: 80 - 40 = 40, 40 - 20 = 20, 20 + 10 = 30, 30 + 15 = 45.
In all cases, Curt has different number of chips at the end (30, 45, 45, and 45) so he could have won or lost chips.