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