Calculate probabilities for coin toss trials, exact heads/tails count, at least k heads, and consecutive streak probabilities.
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The Coin Flip Probability Calculator analyzes independent Bernoulli trials with binary outcomes (Heads vs Tails). It calculates the probability of obtaining exact numbers of heads, cumulative thresholds, and the likelihood of observing long consecutive streaks.
P(X = k) = \binom{n}{k} (0.5)^n, \quad \text{Streak recurrence: } A_m(n) = \sum_{j=1}^m A_m(n-j)Where n is total flips, k is heads observed, and streak probability is solved via Markov state transition recurrence.
Scenario: Find the chance of getting exactly 7 heads when flipping a fair coin 10 times.
n = 10, k = 7, p = 0.5.
Total possible sequences: 2^10 = 1,024.
Number of favorable sequences: C(10, 7) = 120.
P(X = 7) = 120 / 1024 ≈ 0.11719 (11.72%).
The probability of 10 consecutive heads on a fair coin is (1/2)¹⁰ = 1 / 1,024 ≈ 0.0977%, or odds of 1,023 to 1 against.
The Gambler’s Fallacy is believing that after rolling 5 heads in a row, tails is "due" to come up. Because each toss is statistically independent, the probability of heads on the next flip remains exactly 50%.
Human intuition underestimates clustering in random processes. In 200 random flips, a streak of 7 heads or tails in a row occurs more than 75% of the time.
Stanford research led by Persi Diaconis revealed real spun coins show slight physical bias toward the heavier face, and flipped coins land on the face they started on roughly 50.8% of the time.
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