Get Exactly 3 Heads in 5 Coin Flips

Model guide

Why this probability model fits

This is a binomial probability problem because the flips are independent, each flip has the same probability of heads, and the question asks for an exact success count.

Setup: Each flip has the same probability p = 0.5 for heads, and one flip does not change the next one.

Replacement: Replacement is not the right idea here. Coin flips are repeated independent trials, not draws from a shrinking pool.

Independence: Independence matters. The binomial model only fits when one trial does not change the probability of the next one. The outcome of one trial does not affect the outcome of any others.

Worked solution

31.25%

The probability of exactly 3 successes is 31.25%.

1. Identify this as repeated independent trials with the same success probability.
2. Use P(X = k) = C(n, k)p^k(1-p)^(n-k).
3. Set n = 5, target successes = 3, and p = 0.5.
4. Evaluate the required binomial term for the requested success range.
5. The final probability is 5/16, which is 31.25%.