Flip a Coin

Fair coin probability: A standard coin has exactly two outcomes, heads or tails, each with equal probability.

P(Heads) = P(Tails) = 1/2 = 50%

Law of Large Numbers: Over a small number of flips, results can vary wildly from 50/50. Flip 10 coins and you might get 8 heads. But as the number of flips increases, the observed frequency gets closer and closer to the true probability of 50%.

Why do streaks happen? Streaks feel surprising, but they are completely normal. In 100 flips of a fair coin, there is about a 97% chance of seeing a streak of 5 or more in a row. Streaks are not evidence of bias. They are exactly what randomness looks like, and a sequence with no streaks in it would be the suspicious one.

Expected vs. observed frequency: With n flips, the expected number of heads is n/2. The standard deviation of the count is √(n × 0.25) = √n / 2.

Example: 100 flips → expected 50 heads, standard deviation ≈ 5. So getting between 40 and 60 heads is completely normal (within 2 standard deviations).

Binomial distribution: The probability of getting exactly k heads in n flips is: P(k) = C(n,k) × (0.5)^n

where C(n,k) = n! / (k! × (n−k)!) is the number of combinations.

Weighted coin: A biased or weighted coin has P(Heads) ≠ 0.5. Physical coins are very close to fair when flipped. Spinning one on a table is a different matter: the rim is not perfectly symmetrical, and research has found spins favour one side by a few percent. If you are settling something with a coin, flip it rather than spin it.