James and Kate decide to play a simple game of chance. They will take turns
flipping a coin. Each time that it comes up HEADS, James wins; each time that it
comes up TAILS, Kate wins. The first player to win a total of three flips wins the
match.
Assuming that heads and tails are equally likely to be flipped on each toss:
1) What is the probability that the match will end after just three flips?
2) Is the match more likely to end after four flips or after five flips (explain).
Solution to the Problem:
1) The match will end after three flips one-fourth of the time.
The probability of three heads in a row is 1/8,
and the probability of three tails in a row is also 1/8.
There are six possible ways for the match to end after four flips, and six possible ways to produce a 2-2 tie after four flips.
I used a tree diagram to produce all possible games:
H H H
H H T H
H H T T H
H H T T T
H T H H
H T H T H
H T H T T
H T T H H
H T T H T
H T T T
T T T
T T H T
T T H H T
T T H H H
T H T T
T H T H T
T H T H H
T H H T T
T H H T H
T H H H
The probability for each of the possibilities can be figure in the following manner:
If the match ends after three flips, the probability for that match is is 1/2 * 1/2 * 1/2 = 1/8.
If the match ends after four flips, the probability for that match is 1/2 * 1/2 * 1/2 * 1/2 = 1/16.
If the match ends after five flips, the probability for that match is 1/2 * 1/2 * 1/2 * 1/2* 1/2 = 1/32.
Now, you need to find how many matches ended in 3, 4, or 5 flips.
The probability that the match ends in 3 flips is 1/8 + 1/8 = 1/4.
The probability that the match ends in 4 flips is 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 = 3/8.
The probability that the match ends in 5 flips is 1/32 + 1/32 + 1/32 + 1/32 + 1/32 + 1/32 +
1/32 + 1/32 + 1/32 + 1/32 + 1/32 + 1/32 = 3/8.
Correctly solved by:
| 1. David and Judy Dixon | Bennettsville, South Carolina |
| 2. Magdy Essafty | Alexandria, Egypt |
| 3. Luke Mansour |
Brookstone School Columbus, Georgia |
| 4. Richard K. Johnson | La Jolla, California |