How to use this coin flip probability calculator
Enter the number of independent fair-coin flips, choose exactly, at most, or at least, and enter the target number of heads.
- Enter 0–200 flips.
- Choose the event type. “At least” and “at most” include the target.
- Enter heads from 0 through the number of flips.
- Select Calculate and read the percent, decimal, and complement.
- Changing any input invalidates the confirmed result and copy action.
Binomial probability formula
For exactly k heads in n fair flips, P(X = k) = C(n,k) / 2ⁿ. This is the binomial distribution with p = 0.5.
At most k adds the probabilities from 0 through k. At least k adds from k through n. The calculator builds the finite distribution iteratively, which also handles n = 0 cleanly.
Worked examples
Exactly 8 heads in 10 flips
C(10,8) / 2¹⁰ = 45/1024, which is about 4.39%.
At least 8 heads in 10 flips
Add the probabilities for 8, 9, and 10 heads: 56/1024, or about 5.47%.
Zero flips
With no flips, the only possible head count is 0. Exactly 0, at most 0, and at least 0 all have probability 100%.
Assumptions and limits
- Each flip has two outcomes and P(heads) = P(tails) = 0.5.
- Flips are independent; an earlier result does not change the next probability.
- The target is an inclusive whole-number count, not a percentage.
- Theoretical probability does not guarantee a particular short-run result.
- Biased coins and streak order are outside this page.
Frequently asked questions
Does at least 3 include exactly 3?
Yes. It includes 3, 4, and every higher possible head count.
Why can I enter zero flips?
It is a useful boundary case: there is exactly one empty outcome and it contains zero heads.
Can this predict the next flip?
No. For a fair independent coin, the next head probability remains 50%.
Does order matter?
This page counts total heads only. It does not calculate consecutive streaks or a specific sequence.