How Coin Flip Probability Works
Each coin flip is an independent event with two outcomes: heads or tails. For a fair coin, both outcomes have a probability of 0.5 (50%). When you flip a coin n times, the total number of heads follows a binomial distribution. The binomial formula gives the probability of getting exactly k heads in n flips:
P(X = k) = C(n, k) x p^k x (1 - p)^(n - k)
Where C(n, k) is the number of combinations (ways to choose k heads from n flips), p is the probability of heads on each flip, and (1 - p) is the probability of tails.
Worked Example
For 10 fair coin flips, the probability of getting exactly 5 heads is:
- C(10, 5) = 252 (there are 252 ways to arrange 5 heads and 5 tails in 10 flips)
- p^k = 0.5^5 = 0.03125
- (1-p)^(n-k) = 0.5^5 = 0.03125
- P = 252 x 0.03125 x 0.03125 = 252 / 1,024 = 0.2461 = 24.61%
So even though 5 heads is the most likely single result, it only occurs about 1 in 4 times because there are 11 possible outcomes (0 to 10 heads) and many outcomes are close to equally likely.
Cumulative Probabilities
To find the probability of at least k heads, you sum the exact probabilities from k through to n. To find at most k heads, you sum from 0 through to k. These cumulative probabilities are useful for understanding how likely a run of luck is. For example, the probability of getting at least 8 heads in 10 fair flips is P(8) + P(9) + P(10) = 4.39% + 0.98% + 0.10% = 5.47%.
Expected Value and Standard Deviation
The expected number of heads is simply n x p. For 10 fair flips: E = 10 x 0.5 = 5 heads on average. The standard deviation tells you how much the actual result is likely to vary from this average: SD = sqrt(n x p x (1-p)). For 10 fair flips: SD = sqrt(10 x 0.5 x 0.5) = sqrt(2.5) = 1.58. This means a typical result falls within about 1.58 of the average, so roughly 3 to 7 heads is a common range.
Biased Coins
The same formula applies to any fixed probability p. A biased coin with p = 0.6 (60% chance of heads) will on average produce 6 heads in 10 flips. The distribution shifts to the right compared to a fair coin. This calculator lets you set any probability from 0% to 100% to model biased scenarios.
Related Calculators
Method: Binomial distribution formula P(X = k) = C(n,k) x p^k x (1-p)^(n-k). C(n,k) computed using multiplicative formula. Cumulative probabilities summed from the distribution. Expected value E = np; standard deviation SD = sqrt(np(1-p)).
This calculator is for educational purposes. Each coin flip is assumed to be independent with a fixed probability per flip. Real-world coins are not perfectly fair; minor biases in manufacturing or flipping technique can affect outcomes over many trials.
