Coin Flip Probability Calculator

What the coin flip probability calculator does

Flip a coin several times and you can ask how likely a certain number of heads is. This calculator answers that, for exactly a number of heads, at least that many, or at most that many. It works for a fair coin and for a weighted one, since you can set the chance of heads yourself.

Coin flips are the textbook example of the binomial pattern, repeated tries with two outcomes each. Below is how it works and one result that surprises people.

How to use it

1. Enter the number of flips.
2. Choose exactly, at least, or at most, and enter the number of heads.
3. Set the probability of heads, which stays at 0.5 for a fair coin, then press Calculate, or Reset to clear it.

How the probability is worked out

The chance of getting exactly a certain number of heads combines two things: how many ways that number can occur among the flips, and how likely any one such sequence is. The number of ways is a combination, counting which flips landed heads, and each sequence has a probability from multiplying the head and tail chances together:

P(exactly k heads) = (ways to place k heads in n flips) × pⁿ × (1 minus p) to the power of the rest

Here p is the chance of heads. This is the binomial probability formula. It balances the many arrangements that give k heads against the shrinking chance of each specific arrangement, and the result is the probability of that count.

Exactly, at least, and at most

Exactly gives the chance of one specific number of heads, like precisely 5. At least adds up the chances of that number and every larger one, for questions like 5 or more heads. At most adds up that number and every smaller one, for 5 or fewer.

The two cumulative options answer the questions that come up most in practice, since people usually care about a threshold rather than an exact count. Each is just a sum of the exact probabilities across the relevant range, which the calculator handles for you.

Why five heads in ten is not a sure thing

It is tempting to think that because a fair coin averages half heads, ten flips will usually give exactly five. In fact exactly five heads happens only about a quarter of the time. Five is indeed the single most likely count, but it shares the space with four, six, and every other possibility, so no one count claims a majority.

The lesson is that the average outcome is not the guaranteed one, and often not even a likely one on a single try. What is reliable is the pattern across many flips, not the exact result of any one batch. This is the same idea as an expected value being a long-run average rather than a prediction.

A worked example

Suppose you flip a fair coin 10 times and ask for the chance of exactly 5 heads. There are 252 ways to get 5 heads among 10 flips, and each specific sequence has a probability of one in 1,024.

Multiplying gives 252 ÷ 1,024 = about 24.6 percent. So even the most likely single count, 5 out of 10, happens only around a quarter of the time. Ask instead for at least 5 heads, and the chance climbs above half, because it gathers up 5, 6, 7, and all the rest.

Entering your values

Enter the number of flips, the number of heads, and the probability of heads as a decimal between 0 and 1. The calculator returns the probability, the chance as a percentage, and the odds as a "1 in" figure. Because the count of arrangements is built from factorials, which grow very large, extremely high numbers of flips push past what can be computed exactly.

Questions people ask

How do I find the probability of a number of heads?

Multiply the number of ways that many heads can occur by the chance of each sequence. That is the binomial probability, which this calculator computes for you.

What is the difference between exactly, at least, and at most?

Exactly is one specific count. At least sums that count and all larger ones. At most sums that count and all smaller ones.

Can I use it for a weighted coin?

Yes. Set the probability of heads to whatever it is. Leave it at 0.5 for a fair coin, or change it for a biased one.

Why isn't exactly half heads the most likely by a lot?

It is the single most likely count, but it competes with every other possible count. For 10 flips, exactly 5 heads happens only about a quarter of the time.

References

A quick note on where the methods here come from. The binomial probability behind coin flips is set out in the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public statistics reference. OpenStax Introductory Statistics is a free, widely used textbook covering the binomial distribution.

1. NIST/SEMATECH e-Handbook of Statistical Methods (the binomial distribution). https://www.itl.nist.gov/div898/handbook/
2. OpenStax, Introductory Statistics (the binomial distribution). https://openstax.org/details/books/introductory-statistics-2e