Different Number of Heads

Both Bob and Alice toss a fair coin three times and record the results. The probability that Alice records a different number of heads than Bob does can be expressed as $\frac{M}{N}$ where $$ M $$ and $$ N $$ are relatively prime integers. Compute $$ M + N $$ .

Answer

The total number of possible outcomes for both Alice and Bob is:

2^3 \times 2^3 = 64.

Next, calculate the number of outcomes where Alice and Bob have the same number of heads:

0 heads: $1 \times 1 = 1$
1 head: $3 \times 3 = 9$
2 heads: $3 \times 3 = 9$
3 heads: $1 \times 1 = 1$

Total same number of heads:

\[ 1 + 9 + 9 + 1 = 20. \]

Number of different outcomes:

\[ 64 - 20 = 44. \]

Probability of different number of heads:

\frac{44}{64} = \frac{11}{16}.

Thus, $$ M = 11 $$ and $$ N = 16 $$ , so:

\[ M + N = 11 + 16 = 27. \]

Notes and comments

Comment 1: The main point here is that the complement of the event "Different Number of Heads" is "Same Number of Heads".