Table of Contents
Coin Flip Probabilities
The probability of getting exactly k heads in n flips of a coin with success probability p follows the binomial distribution. This is one of the most fundamental probability calculations and forms the basis for statistical hypothesis testing.
For a fair coin (p=0.5), the distribution is symmetric. The most likely outcome is n/2 heads, and the distribution becomes more concentrated around this value as n increases (Central Limit Theorem).
Formula
P(X = k) = C(n,k) × p^k × (1-p)^(n-k)
Expected value = np, Variance = np(1-p)
10 Flips, Fair Coin
| Heads | P(exact) | P(cumulative) |
|---|---|---|
| 0 | 0.0010 | 0.0010 |
| 3 | 0.1172 | 0.1719 |
| 5 | 0.2461 | 0.6230 |
| 7 | 0.1172 | 0.9453 |
| 10 | 0.0010 | 1.0000 |
FAQ
What is the probability of all heads?
P(all heads) = 0.5^n. For 10 flips: 0.5^10 = 0.000977 (about 1 in 1024).
Is "at least k heads" more likely than "exactly k"?
Yes, "at least k" includes all outcomes from k to n, so it is always greater than or equal to "exactly k".