Probability Basics, With Examples

The basic idea

A probability is a number between 0 and 1 that says how likely something is: 0 means it never happens, 1 means it always does, and 0.5 is an even chance. When every outcome is equally likely, you find it by counting: favourable outcomes divided by total outcomes. A fair die has six faces, so the chance of any one number, say a 6, is 1 out of 6, about 16.7%.

AND: independent events multiply

When you want two things to both happen and one does not affect the other, multiply their probabilities. The chance of heads on one coin flip is ½; the chance of heads twice in a row is ½ × ½ = ¼, or 25%. The same logic scales: rolling two sixes with two dice is 1/6 × 1/6 = 1/36, because the dice are independent.

OR: mutually exclusive events add

When you want one of two things to happen and they cannot both occur at once, add their probabilities. On a die, rolling a 1 or a 2 is 1/6 + 1/6 = 2/6 = 1/3. This only works when the events are mutually exclusive; if they can overlap, you subtract the overlap so it isn't counted twice.

The complement

The chance that something does not happen is 1 minus the chance that it does: P(not A) = 1 − P(A). This is the easy way to handle 'at least one' questions. The chance of at least one head in three flips is 1 minus the chance of no heads, 1 − (½)³ = 1 − 1/8 = 7/8, far quicker than adding every winning case.

Frequently asked questions

What's the probability of rolling a 6?

On a fair six-sided die, it's 1 in 6, or about 16.7%.

How do I find the chance of two independent events?

Multiply their individual probabilities. Two ½ events happening together is ½ × ½ = ¼.

What does 'mutually exclusive' mean?

Two events that cannot both happen at the same time, so you can simply add their probabilities.