Comprehensive Calculator Guide
This calculator gives the binomial chance of exactly, at least or at most k heads in n flips. Exactly 5 heads in 10 flips is about 24.6%; 8 or more is about 5.5%. It also shows nearby counts and whether your result is unusual.
The binomial
P(X = k) = C(n, k) p^k (1 − p)^(n − k). A coin has no memory, so it stays a fair way to draw lots; over 100 flips anything from 40 to 60 heads is normal.
How to Use
- Enter the flips n and heads k.
- Change p for a biased coin.
- Press “Calculate” to see exactly, at least and at most.
Formula Used
P(X = k) = C(n, k) × p^k × (1 − p)^(n − k)
Practical Examples
- Example 1
- 5 of 10: 24.61%.
- Example 2
- 8 of 10: 4.39%; at least 8: 5.47%.
Important Tips
- “At least” is often the useful question.
- Test a suspect coin with many flips.
- Works for any two-outcome trial.
Common Mistakes to Avoid
- The gambler’s fallacy.
- Forgetting C(n, k).
- Expecting exactly half.
Frequently Asked Questions
- Chance of heads?
- 50% every flip.
- Exactly vs at least?
- At least includes everything above.
- Is a coin toss fair?
- Yes, with a balanced coin.