Coin Toss Streak Calculator

A streak (or run) in coin flips refers to getting the same result — typically heads — several times in a row. For example, flipping heads three times consecutively is a streak of length 3. Streak probability explores how likely such runs are in a given number of tosses.

The probability is calculated using a recurrence relation based on k-step Fibonacci sequences. For n flips and a desired streak of k, you count the number of sequences of length n that contain (or don't contain) a run of k heads, then divide by the total number of possible sequences (2^n).

A recurrence relation is a formula that defines each term in a sequence using previous terms. For streak calculations, the number of valid sequences of length n without a streak of k heads can be expressed as f(k, n) = f(k, n-1) + f(k, n-2) + … + f(k, n-k), which is a generalization of the Fibonacci sequence.

Out of 8 total outcomes (2^3), the sequences with no two consecutive heads are: TTT, TTH, THT, HTT, HTH — that is 5 sequences. So the probability of no consecutive heads in 3 flips is 5/8 = 62.5%.

Using the recurrence relation, the number of 10-flip sequences with no two consecutive heads is 144 (the 12th Fibonacci number). With 2^10 = 1024 total outcomes, the probability is 144/1024 ≈ 14.06%.

No — by symmetry of a fair coin, the probability of a streak of k heads in n flips is exactly the same as the probability of a streak of k tails. This calculator focuses on heads, but the results apply equally to tails.

'At least k' means the longest streak is k or longer. 'Exactly k' means the longest streak is precisely k (not shorter, not longer). 'At most k' means the longest streak never exceeds k consecutive heads. Each measures a different aspect of streak behavior in a sequence of flips.

The count of flip sequences without a streak of k consecutive heads follows a k-step Fibonacci recurrence — where each term is the sum of the previous k terms. For k=2, this reduces to the standard Fibonacci sequence. This elegant mathematical connection makes streak probabilities computable for any k and n.