Coin Flip Probability: Independence, Streaks, and Exact Odds

A fair coin lands heads with probability 0.5 and tails with 0.5 on every flip. That’s the starting point, but there’s more to understand: independence, why streaks aren’t suspicious, and how to compute exact probabilities for multiple flips.

Flip online and run small experiments with /other/flip-a-coin-simulator.

Independence and Memorylessness

• Each flip is independent of previous results. Ten heads in a row doesn’t make tails “due”—that’s the gambler’s fallacy.
• Short‑run weirdness is normal; long‑run frequencies converge toward 50/50 (law of large numbers).

Exact Odds for Multiple Flips (Binomial)

The number of heads in n flips follows a binomial distribution with p = 0.5.

• P(exactly k heads in n flips) = C(n, k) × (1/2)^n
• Example: In 5 flips, P(exactly 3 heads) = C(5, 3) × (1/32) = 10/32 ≈ 31.25%
• Example: In 10 flips, P(at least 8 heads) = [C(10,8) + C(10,9) + C(10,10)] / 2^10

Streaks Happen—Here’s Why

• In 100 flips, longest runs of 5–7 heads or tails aren’t unusual.
• Humans are pattern‑hungry; clusters look meaningful even when they’re random.

Quick Experiments to Build Intuition

• Flip 50 times; track heads proportion each 10 flips. Watch it wander, then gravitate toward 0.5.
• Run five sets of 20 flips; record the longest streak each set—compare across sets.

Real‑World Imperfections

• Minor biases can arise from coin damage or technique, but consistent 50/50 behavior is typical with clean coins and consistent flips. For strict fairness, use the simulator to remove human variation.

FAQs

Does a streak make the opposite outcome more likely next time? No. Independence means each flip is still 50/50 regardless of previous flips.

How do I calculate “at least” probabilities? Sum the exact‑k probabilities or use a binomial calculator for speed.