Coin Toss Streak Calculator
Calculate the probability of consecutive heads or tails in coin flip sequences
Calculate Streak Probability
Total number of coin tosses (1-100)
Number of consecutive heads/tails
Type of streak probability to calculate
Display format for probability
Streak Probability Results
Selected Result
Mathematical Calculation
Formula used: P(L ≤ k) = f(k,n) / 2^n
Recurrence relation: f(k,n) = Σ f(k, n-i) for i = 1 to k+1
Total possible outcomes: 2^10 = 1,024
Sequence type: 4-step Fibonacci (Tetranacci)
Probability Interpretation
4-step Fibonacci Sequence
The f(k,n) values used in calculating streak probabilities follow a 4-step Fibonacci pattern:
Example: 3 Heads in 10 Flips
Problem Setup
Question: What's the probability of getting at least 3 consecutive heads in 10 coin flips?
Method: Use f(k,n) recurrence relation
Solution
P(L ≥ 3) = 1 - P(L ≤ 2)
P(L ≤ 2) = f(2,10) / 2^10
f(2,10) = 504 (Tribonacci sequence)
Result: ≈ 50.78%
Fibonacci Sequences
Statistical Tips
Longer streaks become exponentially less likely
Order matters: HHHTT ≠ HTHTH despite same totals
At-least probabilities are always ≥ exact probabilities
More flips increase chances of longer streaks
Understanding Coin Flip Streak Probabilities
What Are Streaks in Coin Flips?
A streak (or run) occurs when you get the same result several times in a row. Unlike basic coin flip probability where only the total count matters, streak analysis considers the order of results. For example, HHHTH has a streak of 3 heads, while HHTHH has only a streak of 2 heads.
Why Study Streaks?
- •Understanding randomness and probability patterns
- •Applications in gambling and game theory
- •Quality control and statistical process control
- •Financial market analysis and trend detection
Mathematical Framework
P(L ≤ k) = f(k,n) / 2^n
f(k,n) = Σ f(k, n-i) for i = 1 to k+1
- L: Length of longest head streak
- k: Target streak length
- n: Total number of coin flips
- f(k,n): Number of sequences with max streak ≤ k
Key Insight: The f(k,n) values follow generalized Fibonacci sequences, making this problem connect to famous mathematical sequences!
Probability Types Explained
Exactly k
The probability that the longest streak is exactly k heads in a row. Calculated as P(L ≤ k) - P(L ≤ k-1).
At Least k
The probability of getting k or more consecutive heads. Calculated as 1 - P(L ≤ k-1).
At Most k
The probability that the longest streak is k heads or fewer. Directly calculated as P(L ≤ k) = f(k,n) / 2^n.