AP Statistics Chapter 15:

Independent doesn't equal Disjoint

Disjoint events can't be independent. SInce we know that disjoint evens have not oucomes in common, knowing that one occurred means the other didn't. This, the probability of the second event occurring changed based on our knowledge that the first event occurred. It follows, then, that the two events aren't independent.

Events A and B are independent whenever

P(B|A) = P(B) or P(A|B) = P(A)

Independence

The outcome of one event doesn't influence the probability of the other

Tree Diagram

Multiply the probabilities of the branches together. All the final outcomes are disjoint and must add up to one. We can add the final probabilities to find probabilities of compound events

Drawing/Sampling Without Replacement

Once one individual is drawn it doesn't go back into the pool. Assume that you pull out your desirable result. Another instance of working with conditional probabilities

Conditional Probability

A probability that takes into account a given condition

The General Addition Rule for Two Events

P(A or B) = P(A) + P(B) - P(A and B)

The General Multiplication Rule

Rearranging the equation in the definition for conditional probability. For any two events A and B, P(A and B) = P(A) X P(B|A) or P(A or B) = P(B) X P(A|B)

To find the probability of event B given the event A

Restrict our attention to the outcomes in A. P(B|A) = P(A and B)/ P(A). P(A) can't equal 0, since we know that A has occurred

The General Addition Rule

When two events A and B are disjoint, we can use the addition rule for disjoint events: P(A or B) = P(A) + P(B). When events aren't disjoint, this earlier addition rule will double the probability of both A and B occurring.

Multiplication Rule

When two events A and B are independent, we can use the multiplication rule for independent events. P(A and B) = P(A) X P(B). When events aren't independent, this earlier multiplication rule doesn't work