How to Calculate the Probability of Independent Events

Dice are an often used synecdoche for explaining statistics. Rolling multiple dice is tantamount to calculating the probability of two independent events occurring together. To state it generally, the probability of two events occurring together is equal to the product of the probabilities of each event occurring alone. To state it with an equation for two events A and B: p(AB) = p(A) * p(B). This applies to any number of events that one looks at.

Statistical independence can be a complicated phenomenon. Simple events such as flipping a coin or rolling dice is almost always independent events because the outcome you got on one roll does not effect the probability of what you get on any other roll. Let’s consider an example of a statistically dependent event. If you were to ask people on the street who is the current director of the White House Office of Science and Technology Policy (it’s John Holdren), the first person you ask might answer Joe Biden (obviously not, he’s the vice-president). Say 3 of 4 people nearby on the street do not know the answer either, and when asked answer the same. You would measure that 80% of your sample answered Joe Biden, but their answers were clearly not influenced!

To state some justification for our statement p(AB) = p(A) * p(B), let us consider the probability of rolling double sixes with two six-sided gaming dice. We roll the two dice, and look at the outcomes one by one. There is a 1 in 6 probability of rolling a six on the first dice. Looking at the second dice, there is also a 1 in 6 probability of having rolled a six on the second dice. Therefore there are 36 possible outcomes, of which only 1 is double sixes, therefore the probability of rolling double sixes is 1/36 or just under 3% of the time.

A useful device to visualize the math behind simple events like this is to think of a multilevel tree: at level one, there are 6 branches, one for each outcome. Branching off from each of those are 6 additional branches, one for each of the outcomes of the second dice. At the level with the most branches (representing the final outcome) there are a total of 36 branches, so each outcome has a 1 in 36 chance of happening.

We can evaluate the probability of any arbitrary combination occurring by summing the probability of each outcome which gives us the desired results. For example, rolling any double, rather than just double sixes has 6 possible combinations (one for each number) so the probability is 1/36+1/36+1/36+1/36+1/36+1/36 = 1/6 or about 17% of the time. This seems a bit redundant when each probability has the same chance of occurring, but works in any general case.

These methods can easily be extended to any number of dice or events. To recap, the probability of any combination happening is the product of the probability of each event occurring and the probability of any combination is the sum of the probability of any outcome which fits your criteria. Enjoy your exploration of probability and statistics!