# How to calculate the probability of a set of events ?

Everyone knows that the probability of an outcome of heads in a single coin toss is 50% and that each toss is a completely independent event from previous and future tosses. But what about a set of consecutive identical outcomes (i.e. three or four or five, etc. heads in a row) ? How can I calculate the probability of a set of three consecutive heads (my desired outcome) in a series of 100 coin tosses ? Or a set of four consecutive heads in a series of 500 coin tosses ?

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## 4 Answers

The probability of each outcome in a coin toss is 50% (½) because there are only two possibilities. How many possibilities are there if we toss the coin twice? Four. It can be HH, HT, TH, or TT. So the chance of getting HH is 25% (¼), or ½×½. In other words, it’s the probability of getting heads on the first toss multiplied by the probability of getting heads on the second toss. And it keeps going this way. If you toss a coin 3 times, there are eight possibilities: HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT. That’s ½×½×½ = 12.5%.

Assuming you do not want to work out all the possible outcomes for 100 or 500 coin tosses, however, you’ll need a formula for determining the probability of getting at least one run of *x* heads in a series of *y* tosses. There are at least three available options. Unfortunately, they all involve rather complicated math. You can find a discussion of them here (search “non-recursive” for the first two and “Fibonacci” for the third, which is also the last on the page).

What you are describing is often called a *run*. Here is a Ask a Mathematician thread on this topic.

The exact solution is *very* ugly and complicated, but a rough approximation is:

E = N * q * p^k

Where:

p = probability of heads

q = 1 – p = probability of tails

k = length of run

N = number of flips, which should be *much greater* than k

E = expected number of runs of length k in a sample of size N

edit: you may notice that this does not work when, say, the value of q is very large or small. This is part of being an approximation.

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