Breaking Even
Alice and Bob each start with $4 and bet on a series of 8 successive coin tosses. For each toss, if the coin lands on heads, Alice gives Bob $1; if it lands on tails, Bob gives Alice $1. What is the probability that they break even after the 8 rounds, with neither of them ever running out of money?
Answer
To break even after 8 tosses, there must be exactly 4 heads and 4 tails. The number of ways to achieve this is:
\[
\binom{8}{4} = 70.
\]
However, we must exclude the two outcomes that lead to bankruptcy:
\[
\text{HHHHTTTT and TTTTHHHH}.
\]
Thus, the number of favorable outcomes is:
\[
70 - 2 = 68.
\]
The total number of possible outcomes when flipping a coin 8 times is:
\[
2^8 = 256.
\]
Therefore, the probability of breaking even without bankruptcy is:
\[
\frac{68}{256} = \frac{17}{64}.
\]