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 use the AND Probability Calculator.
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. You might also find our use the Subset Calculator useful.
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'.