Coin Toss Streak Calculator
Coin toss streak calculator. P(streak ≥ s) in n flips, expected longest run ≈ log₂(n). Heads, tails,
Why This Statistical Analysis Matters
Why: Statistical calculator for analysis.
How: Enter inputs and compute results.
Coin Toss Streak — How Common Are Long Runs?
Calculate P(streak ≥ s) in n flips. Expected longest streak ≈ log₂(n) for fair coins. Streaks are more common than you think.
📊 Quick Examples — Click to Load
Inputs
Streak Probability Table (s = 1 to 15)
| s | P(streak ≥ s) |
|---|---|
| 1 | 100.0000% |
| 2 | 100.0000% |
| 3 | 99.9738% |
| 4 | 97.2715% |
| 5 | 81.0110% |
| 6 | 54.6094% |
| 7 | 31.7520% |
| 8 | 17.0208% |
| 9 | 8.7559% |
| 10 | 4.4137% |
| 11 | 2.2029% |
| 12 | 1.0942% |
| 13 | 0.5422% |
| 14 | 0.2683% |
| 15 | 0.1327% |
Streak Probability vs Length
P(streak ≥ s) for s = 1 to min(20, n). Current s highlighted.
Streak Probability vs Flips
How probability grows with n for fixed s = 5.
For educational and informational purposes only. Verify with a qualified professional.
Key Takeaways
- Streaks are more common than people think — in 100 fair coin flips, expect a streak of ~7 consecutive heads
- P(5 heads in a row within 20 flips) ≈ 25% — much higher than most people guess
- The expected longest run of heads in n flips ≈ log₂(n) for a fair coin
- Streaks do NOT indicate a "hot hand" or pattern — they're a natural consequence of randomness
- The gambler's fallacy: past flips don't affect future ones — each flip is independent
Did You Know?
How It Works
1. Markov Chain / DP
We track the probability of being at streak length j after each flip. Once we reach s, we "absorb" (streak achieved).
2. State Transitions
Heads: extend streak (j → j+1). Tails: reset streak (j → 0). P(streak) = 1 − P(never reached s).
3. Expected Longest Streak
For fair coin: E(longest run) ≈ log₂(n). For biased coin: E ≈ log(n) / log(1/p).
4. Either Streak
P(either heads OR tails streak) ≈ 1 − (1−P_H)(1−P_T). Approximate when treating the two events as independent.
Expert Tips
Streaks Are Common
In 20 flips, 5 heads in a row happens ~25% of the time. Don't be surprised.
Expected vs Actual
Expected longest streak is an average — actual runs vary widely.
Biased Coins
Use p ≠ 0.5 for weighted coins, free throws, or sports win rates.
Gambler's Fallacy
After 5 heads, P(next heads) is still 50%. Past flips don't affect future ones.
Comparison Table
| Feature | This Calculator | Simulation | Manual |
|---|---|---|---|
| Exact streak probability (DP) | ✅ | ⚠️ Approximate | ⚠️ Complex |
| Expected longest streak | ✅ | ⚠️ Empirical | ✅ |
| Heads / Tails / Either | ✅ | ✅ | ⚠️ Manual |
| Charts (vs length, vs flips) | ✅ | ❌ | ❌ |
| Educational content | ✅ | ❌ | ❌ |
Frequently Asked Questions
What is the probability of 5 heads in a row within 20 flips?
P ≈ 25% for a fair coin. Most people guess much lower. Streaks are more common than intuition suggests.
What is the expected longest run of heads in 100 flips?
For a fair coin: E ≈ log₂(100) ≈ 6.6. So you expect a streak of about 7 heads somewhere in 100 flips.
Does a streak of heads mean tails is "due"?
No. The gambler's fallacy: each flip is independent. P(heads) = 50% regardless of past outcomes.
What is P(10 heads in a row) exactly?
P(10 heads in a row at any point) = 1/1024 ≈ 0.098% for a single sequence of 10. Within n flips, use the DP formula.
How does biased coin affect streaks?
Higher p (heads) = more head streaks. Expected longest streak ≈ log(n) / log(1/p).
What is "Either" streak?
Probability of getting s consecutive heads OR s consecutive tails at some point. Higher than heads-only.
Can I use this for free throws or sports?
Yes. Use p = success rate (e.g., 0.85 for 85% free throw). Same math applies.
Why is the expected streak log₂(n)?
From Feller and Schilling: the longest run grows logarithmically with n. For fair coin, base ≈ 2.
Infographic Stats
Official Sources
Disclaimer: This calculator provides coin toss streak probabilities for educational and reference purposes. The formula assumes independent flips with constant probability p. For critical applications, verify results against established statistical software. The "Either" streak uses an approximation.
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