CHAPTER 5 SUMMARY Probability Rules: Probability of an Outcome A (or Event A) – denoted as P(A) – is the likelihood that A will occur. Sample Space is the set of all possible outcomes. The probability of an event that could not possibly occur is 0 and the probability that an event will certainly occur is 1. So for any probability 0 ≤ P(a) ≤ 1. The sum of the probabilities of all possible outcomes in the sample space is 1. Two methods of finding a probability of an event: Frequency ( A) where n is the number of trials. n #_ of _ ways _ A _ can _ occur Classical – P( A) . #_ of _ all _ possible _ outcomes Empirical (or Experimental) – P( A) The empirical method is an estimate of the true probability which is the determined with the classical method. Law of Large Numbers – As the number of trials (n) increases P(A) from the empirical method approaches the P(A) of the classical method. Mutually Exclusive Events and the Addition Rule and Compliments: If two events, A and B, can NOT occur at the same time they are said to be Mutually Exclusive (or Disjoint). If two events, A and B, are mutually exclusive then P A B P( A) P( B) . If two events, A and B, are NOT mutually exclusive then P A B P( A) P( B) P A B . P A 1 P( A) where P A means the probability of A not occurring. Independence and the Multiplication Rule and Conditional Probability: Events A & B are said to be Independent if the occurrence of A does not effect the probability of B occurring. P B | A is the probability of B occurring given that A has already occurred. If two events, A and B, are independent then P A B P( A)* P( B) . If two events, A and B, are NOT independent then P A B P( A)* P B | A . Counting Events: Multiplication Rule: If there are several choices to make (p being the number of the first choices, q being the number of second choices and r being the number of third choices) then the total number of all choices is p*q*r. Counting With Replacement: If you are selecting from n items and you can reuse the items, then you have n choices for the first selection. Since you can reuse the item you have n choices for the second choice. The same is true for all subsequent choices. If you are making a total of r choices, then the total number of ways r items can be selected from n items is n r . Counting Without Replacement: If you have n choices for the first event but can not reuse that choice, then you only have n-1 choices for the second event and n-2 for the third, and so on. If you are selecting n choices from n items then the total ways they can be selected is n*(n-1)*(n-2)*……..*2*1 or n!. Permutations: If you are making r choices from n items and you can not replace items and the order of the items selected matters, then the total ways r items can be selected is denoted as n Pr which is n! . (n r )! Combinations: If you are making r choices from n items and you can not replace items and the order of the items selected does not matter, then the total ways r items can be selected is denoted as n Cr which is n! . (n r )!r !