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.
2) The chances are equal.
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.