Coin-Flip Probability Calculator
Calculate exact binomial probabilities for a chosen number of coin flips. Your work stays on this device whenever possible.
How to use Coin-Flip Probability Calculator
- Set 'Number of flips (n)' to the total number of independent trials, from 1 to 1000.
- Type the target count into 'Number of heads (k)'; it must be a whole number between 0 and n.
- In the 'Probability' selector pick 'Exactly k heads', 'At least k heads' or 'At most k heads'.
- Leave 'Head probability' at 0.5 for a fair coin, or enter another chance between 0 and 1 for a biased event.
- Press 'Calculate probability' and read the percentage, then the formula line underneath it.
Worked example
Set n = 10, k = 6, p = 0.5 and choose 'Exactly k heads'. The formula is C(10,6) × 0.5^6 × 0.5^4 = 210 ÷ 1024 = 0.205078125, so the tool prints about 20.5078125%. Switching to 'At least k heads' adds C(10,7)=120, C(10,8)=45, C(10,9)=10 and C(10,10)=1 to the 210, giving 386 ÷ 1024 = 37.6953125%.
When to use this tool
- A quality inspector estimating the chance that 3 or more of 20 sampled parts are faulty when the defect rate is 5 percent.
- A game designer checking how likely a 7-of-10 streak is with a slightly weighted die.
Important limitations
The binomial model assumes independent trials with one constant success probability. Biased or dependent real-world events require a different model.
Frequently asked questions
Is this only for coins?
The labels say flips, but the math is the general binomial rule, so any fixed number of independent yes/no trials with one constant chance works. Set 'Head probability' to 0.05 to model a 5 percent defect rate on a production line.
What are the input limits?
n must be a whole number from 1 to 1000, k a whole number from 0 to n, and the probability from 0 to 1. Anything else triggers a single error line and no result is drawn.
How do 'At least' and 'At most' differ?
At least sums the exact probabilities from k up to n; at most sums from 0 up to k. They share the k term, so 'at least 6' plus 'at most 5' equals 100 percent for the same n and p.
How accurate is it for large n?
Each term is built from logarithms, avoiding overflow from values like 1000 choose 500, and shown to ten significant digits. Terms smaller than about 1e-300 underflow to zero, so extreme tails read as 0.
Does it give a full distribution table?
It doesn't print every P(X=k) as a table, but it does show a shaded PMF chart across k and computes mean, variance and standard deviation for you automatically.