Conditional probability and independence

Calculate probabilities with a condition and distinguish independence from mutual exclusivity.

Meaning

Conditional probability measures an event within the outcomes allowed by given information. When Outcomes outside B are excluded, and dividing by P(B) rescales the probability of all of B to 1.Events are independent when knowing that one occurred does not change the other's probability. For check ; in general, independence is expressed by

Example

For a fair die, let A mean a result greater than 4 and B an even result. Among 2, 4, and 6 in B, only 6 also belongs to A, so Without the condition, A contains 5 and 6, so as well. These events are independent.If A instead means rolling 6, its unconditional probability is and its probability given B is The information changes the probability, so these events are not independent.The conditioning event determines the denominator. The conditional-probability formula above cannot use an event of probability zero as its condition. Mutually exclusive events with positive probabilities are not independent: one occurring makes the other impossible.

Practice for this concept

Scope and references

Topics

Probabilities under conditions, independence and dependence, independence versus mutual exclusivity

Learning goal

Calculate probabilities with a condition and distinguish independence from mutual exclusivity.

Helpful background

Check how a condition changes the relevant events and sample space.