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- Question: A science educator is designing a virtual lab where students simulate flipping 6 fair coins and rolling a single 6-sided die. What is the probability that the number of heads equals the value rolled on the die?
- Solution: Let $ H $ be the number of heads in 6 flips of a fair coin, so $ H \sim \text{Binomial}(6, \frac{1}{2}) $. Let $ D $ be the outcome of a fair 6-sided die, uniformly distributed over $ \{1,2,3,4,5,6\} $. We seek $ P(H = D) $.
- Since $ D $ takes integer values from 1 to 6, we only consider values of $ H = k $ for $ k = 1,2,3,4,5,6 $. But $ H $ max is 6, so all values are possible.
- P(H = D) = \sum_{k=1}^{6} P(H = k) \cdot P(D = k) = \sum_{k=1}^{6} \binom{6}{k} \left(\frac{1}{2}\right)^6 \cdot \frac{1}{6}.
- Factor out constants:
- P = \frac{1}{6 \cdot 2^6} \sum_{k=1}^{6} \binom{6}{k} = \frac{1}{384} \left( \sum_{k=0}^{6} \binom{6}{k} - \binom{6}{0} \right) = \frac{1}{384} (2^6 - 1) = \frac{63}{384}.
