Probability Rules

Algebra 2 · Axiom Academy

The complement, addition, and multiplication rules — each shown as the picture that makes it obvious. Every possible outcome together makes up the sample space , and its total probability is exactly 1. So the chance that event A does not happen is simply everything left over once you take A out. Split the sample space into two pieces — A, and everything that is not A. Because the two pieces fill the whole space, their probabilities must add up to 1. Let A be "roll a 1 or a 2", so P(A) = 2/6. The complement is "roll a 3, 4, 5, or 6". For the probability of event A or event B, you add their probabilities — but if the events can happen at the same time, that overlap gets added twice. Subtract it once to fix the double count. In the Venn diagram, P(A) covers the whole left circle and P(B) covers the whole right circle. The lens where they overlap sits inside both, so adding the two circles counts that lens twice — hence the minus. Example: mutually exclusive events On one die roll, let A be "roll a 5" and B be "roll an even number". A 5 is never even, so the events cannot overlap and P(A and B) = 0. 3. The Multiplication Rule (AND) For the probability of A and B both happening, follow a path through a tree and multiply the probabilities along the way. The second factor is a conditional probability, P(B given A) — the chance of B once A has happened. When the events are independent, A tells you nothing about B, so that factor is just P(B).

This is the written version of the interactive lesson above. See the full Algebra 2 course.