What is the probability of getting exactly 5 heads in 10 coin flips?

The probability of getting exactly 5 heads in 10 fair coin flips is approximately 24.61%. This is calculated using the binomial formula: P = C(10,5) x 0.5^10 = 252 / 1024 = 0.2461. Although 5 heads out of 10 is the single most likely outcome, it still only happens about one in four times because there are so many other possible results.

How do you calculate the probability of getting at least k heads in n flips?

To find the probability of at least k heads, sum the binomial probabilities for every outcome from k heads up to n heads. For example, the probability of at least 7 heads in 10 flips equals P(7) + P(8) + P(9) + P(10), where each P(x) = C(10,x) x 0.5^10. This gives approximately 17.19%. The at-most probability works the same way, summing from 0 heads up to k heads.

Does the coin need to be fair for this calculator to apply?

No. This calculator lets you set any probability of heads between 0% and 100%, so you can model biased coins as well as fair ones. A fair coin has a 50% probability of heads on each flip. A biased coin might have a 60% or 30% probability. The binomial formula P(X=k) = C(n,k) x p^k x (1-p)^(n-k) works for any fixed probability p as long as each flip is independent.

Coin Flip Probability Calculator

This calculator works out the probability of getting a chosen number of heads or tails across any number of coin flips, using the binomial distribution formula that underpins probability and statistics. You enter the number of flips (n), the number of heads to check (k), and the probability of heads per flip, which defaults to 50% for a fair coin but can be set from 0% to 100% to model a biased coin, using the built-in presets or your own figure. You then pick a scenario: exactly k heads, at least k heads, at most k heads, fewer than k heads, or more than k heads. The calculator returns the probability of that outcome as a percentage, the equivalent 1 in X chance, the expected number of heads, and the standard deviation, showing how far the actual result typically strays from that average. Below the results sits a calculation breakdown showing the combinations C(n,k) used in the formula, a bar chart of probabilities across every possible number of heads, and a full table listing the exact, at-most, and at-least chance for each outcome from 0 to n. A worked example explains the result in plain language. Use it to check your intuition about coin flip odds, work through probability homework, or see how the chance of an exact outcome falls as flips increase.

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Verified 2025 Binomial distribution formula. Each flip is independent.

1. Flip Settings

Number of flips (n)

Number of heads to check (k)

Probability of heads per flip (%)

2. Scenario

Calculate probability of

Coin type (preset)

Coin Flip Probability Results

Probability
-
of the selected outcome

1 in X chance

approximately

Expected heads

on average (n x p)

Std deviation

spread of outcomes

Calculation Breakdown

Number of flips (n)-

Heads to check (k)-

P(heads) per flip-

P(tails) per flip-

Scenario-

Combinations C(n,k)-

Result probability-

Full Distribution

Worked example: Adjust the inputs above to see your result.

All Possible Outcomes (full probability table)

Heads (k)	Exact probability	Percentage	At most k	At least k

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

Maths and Stats Calculators: browse the full maths collection.
Probability Calculator: general probability and event combinations.
Combination (nCr) Calculator: count ways to choose k from n.
Permutation (nPr) Calculator: ordered arrangements of items.
Bayes Theorem Calculator: update probabilities with new evidence.

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.