This page is about streaks–things like streaks of successful three-point shots in basketball, or consecutive heads when flipping a coin. People are notoriously bad at estimating probabilities in general, and when it comes streaks it’s even worse.
For example, if a basketball player makes several three-point shots in the row, the sportscasters, fans, other players, and even the coach might infer that she’s (or he’s) on a “hot streak,” and think they should “get her the ball.” The notion is that a player on a “streak” is more likely to make that next three-point shot.
This is, of course, nonsense, which is what this page is about.
More generally, this page calculates the probability of observing a minimum of K consecutive successes in N binomial trials, where p is the probability of success.
For a specific example, how likely is it that in 100 flips of a fair coin, you’ll see at least one streak of six consecutive heads?
This is a more complex question than it first appears because there are so many ways this can happen. You might see exactly one such streak, more than one, or you might see one or more streaks of lengths of seven, or eight, or more. Each are among the ways to qualify seeing “at least one streak of at least six in 100 tosses.” The multiplicity of ways that this can happen makes the computations devilishly difficult.
To find out the answer to the question, press “Calculate” in the table below. You might be surprised by the answer.
You can play with the numbers in the table to calculate, for example, the odds of rolling doubles 20 consecutive times out of 500 rolls with fair dice by changing the parameters in the table below, where the probabiity of success is .167, or 1/6.
This uses an elegant recursion formula described on this website. Scroll down to the “mathematician’s answer” for his derivation. Alas, I don’t know his name, so I can’t give him more proper credit. I inferred which pronoun to use from his photo on the above site, which shows he’s male. Anyway, thanks to him and the discussion on the above for motivating this page.