Probability of Streaks Calculator

Understanding Streaks and the Gambler's Fallacy

Streaks are sequences of consecutive outcomes—like getting 5 heads in a row, or losing 10 times consecutively at blackjack. They occur naturally in all random processes, yet they're widely misunderstood and exploited by casinos.

The Formula

The probability of a streak of length n is simply:

Where p is the probability of the individual outcome. For a win streak, use the win probability. For a lose streak, use 1 minus the win probability.

Example: Fair Coin Flips

What's the probability of getting 5 heads in a row?

About 1 in 32. If you flip a coin 32 times, you'll likely see this streak at least once—and sometimes more! This isn't unusual; it's mathematically expected.

Example: Roulette Losing Streak

You're betting red on a European roulette wheel (48.65% win chance). What's the probability of losing 7 times in a row?

Less than 1%, but over 1,000 trials (about 2-3 hours of play), you'd expect to see this happen roughly 9-10 times. It's rare for any specific sequence of 7 spins, but quite likely over an evening of gambling.

The Gambler's Fallacy Explained

The Gambler's Fallacy is the mistaken belief that past outcomes affect future probabilities in independent events. Common forms include:

• "I've lost 10 times in a row, so I'm due for a win." Wrong. If each event is independent, the probability of winning on the next trial is exactly the same as it was before. The outcomes don't "balance out" in the short term.
• "Red has come up 8 times in a row, so black is more likely now." Wrong. Each roulette spin is independent. The probability of red on the next spin is still 48.65%, exactly the same as always.
• "This slot machine hasn't paid out in a while, so it's due to hit soon." Wrong. Modern slot machines use Random Number Generators that ensure each spin is independent. The machine has no memory of previous spins.

Why the Fallacy Persists

Humans are pattern-seeking creatures. We evolved to find cause and effect in our environment, which served us well for survival. But this instinct misleads us with truly random processes:

• Misunderstanding the Law of Large Numbers: Over millions of trials, outcomes do balance out to their expected probabilities. But this doesn't mean short-term imbalances "correct themselves." A streak doesn't make the opposite outcome more likely; it just becomes statistically insignificant as the sample size grows.
• Selective Memory: We remember when we were "right" about a streak ending, but forget the many times it didn't. This confirmation bias reinforces the fallacy.
• Representativeness Heuristic: A sequence like HTHTHTHT "looks" more random than HHHHHHHH, even though both are equally likely for 8 coin flips (1/256 each). We expect randomness to "look random" at every scale, but true randomness includes streaks.

Real-World Costs of the Fallacy

The Gambler's Fallacy has devastating financial consequences:

• Martingale Systems: Doubling your bet after each loss, believing a win is "due," leads to exponential losses. A 10-loss streak (not uncommon) turns a $10 bet into a $10,240 bet.
• Chasing Losses: After a losing streak, players increase bets to "recoup" losses quickly, often leading to even greater losses.
• Extended Play: Believing luck will "turn around," players continue gambling far longer than they planned, compounding negative expected value.

Streaks in Different Games

Different casino games produce different streak frequencies:

• Roulette: With 18/37 ≈ 48.65% for red/black, 5-loss streaks occur about once every 35 spins. 10-loss streaks about once every 1,200 spins.
• Blackjack: With optimal strategy, win rate is about 43%, push 9%, lose 48%. Losing streaks are common due to the slight house edge.
• Slots: Win rates vary (10-25%), so losing streaks dominate. A 10% hit rate means losing 10+ times in a row happens roughly once every 35 spins.
• Craps (Pass Line): 49.3% win rate. Very close to even money, so streaks of both types are common and roughly balanced.

The Correct Interpretation

When you see a streak, the mathematically sound interpretation is:

• "This is a rare but expected occurrence in a random sequence. It neither predicts nor influences future outcomes."
• "Each new event has the same probability as before, regardless of what happened previously."
• "If I continue playing, I will see more streaks of various lengths. Some will favor me, some won't. Over time, expected value determines my result."