How do you calculate coin flip probability?
Coin flip probability uses the binomial probability formula: P(X) = C(n, X) × p^X × (1−p)^(n−X), where n is the number of flips, X is the desired number of heads, and p is the probability of heads on a single flip. C(n, X) is the binomial coefficient, also written as n! / (X! × (n−X)!). For a fair coin, p = 0.5. See also our calculate Posterior Probability Posterior Probability P(H|E).
What is the formula for probability in a coin toss?
The classical probability formula is P(event) = (number of favorable outcomes) / (number of all possible outcomes). For a single fair coin flip, there are 2 possible outcomes, so P(heads) = 1/2 = 0.5. For multiple flips, the binomial distribution formula is used to account for all possible combinations.
How do I compute the probability of 8 heads in 10 tosses?
Set n = 10, X = 8, and p = 0.5, then select 'Exactly X heads'. The binomial formula gives P(8) = C(10, 8) × 0.5^8 × 0.5^2 = 45 × (1/1024) ≈ 0.0439, or about 4.39%. This is much lower than 50% because you're specifying an exact count out of 10 flips.
What is the probability of 2 heads in 3 tosses?
With n = 3, X = 2, and p = 0.5, P(exactly 2 heads) = C(3, 2) × 0.5^2 × 0.5^1 = 3 × 0.25 × 0.5 = 0.375, or 37.5%. There are 3 favorable outcomes (HHT, HTH, THH) out of 8 total possible outcomes.
What is the probability of at least 1 head in 4 tosses?
The easiest approach is to use the complement rule: P(at least 1 head) = 1 − P(0 heads). P(0 heads) = 0.5^4 = 0.0625. So P(at least 1 head) = 1 − 0.0625 = 0.9375, or 93.75%. You can also enter n = 4, X = 1, p = 0.5 and select 'At least X heads' in this calculator.
What is the difference between 'at least X', 'at most X', and 'exactly X' heads?
'Exactly X heads' gives the probability of getting precisely X heads. 'At least X heads' sums the probabilities for X, X+1, …, n heads. 'At most X heads' sums probabilities for 0, 1, …, X heads. 'More than X' and 'Less than X' exclude the boundary value X from those sums respectively.
Does the calculator work for biased (unfair) coins?
Yes. The probability of heads (p) field can be set to any value between 0 and 1. A fair coin uses p = 0.5, but if a coin lands heads 70% of the time, set p = 0.7. The binomial formula still applies — it uses your specified p value for all calculations.
Why isn't the probability of getting exactly 8 heads on 16 flips equal to 50%?
While 8 heads is the most likely single outcome for 16 flips of a fair coin, it is still just one specific outcome among many. The probability of exactly 8 heads is about 19.64%. The remaining ~80.36% is spread across all other possible outcomes (0 through 16 heads, excluding 8). 'Most likely' does not mean 'more likely than not'. You might also find our calculate Permutation and Combination Permutations nPr (Order Matters) useful.