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Unit 3 Chapter 3 - Definitions Probability is a numerical measure between 0 and 1 that describes the likelihood that an event will occur. 0 P(A) 1 never negative or over 1 Notation: P(A) is read as “the probability of event A happening” Event: an event is a collection or set of one or more simple event in the sample space. Sample space: this is the list of all possible outcomes from a random experiment. Pr obability of an event Number of outcomes favorable to event Total number of outcomes The complement of a event A is the event that A does not occur, A ’ 1) P(A) + P(A’ ) = 1 2) P(A ’) = 1 – P(A) and P(A) = 1 – P(A ’ ) Two events are independent if the occurrence or nonoccurrence of one does not change the probability that the other will occur. If events A and B are independent, then P(A and B) = P(A) P(B). Two events are dependent if the occurrence or nonoccurrence of one does change the probability that the other will occur. For dependent events: P(A and B) = P(A) P(B| A) P(B | A) is called a conditional probability because it depends A occurring first. “the probability that event B happens, given that event A has already occurred” Two events are mutually exclusive or disjoint if they cannot occur at the same time. If P(A and B) = 0, the events A and B are mutually exclusive. Addition rule of mutually exclusive events A and B: P(A or B) = P(A) + P(B) Addition rule for not mutually exclusive events A and B: P(A or B) = P(A) + P(B) – P(A and B) Counting techniques Creating groups of characters – this is really an example where the multiplication rule can be applied. Think of the experiment where you are picking numbers out of a hat to fill each place value. How many groups can be created from 1) 4 single digits (recall there are 10 single digits 0 – 9) 10101010 = 10,000 groups 2) 4 single digit without replacement 10987 = 5040 groups 3) 3 single digits followed by 2 letters (recall there are 26 letters) 1010102626 = 676,000 groups Factorial notation For a counting number n 0! = 1 Be very careful with this one!! 1! = 1 n! = n(n – 1)(n – 2)…1 Example: 6! = 6 * 5 * 4 * 3 * 2 * 1 = 720 Counting rule for permutations (Use this rule when it MATTERS which order the objects are in) The number of ways to arrange in order n distinct objects, take then r at a time is n Pr n! (n r )! where n and r are whole numbers and n r . Counting rule for combinations (Use this rule when it DOES NOT MATTER in which order the objects are placed) The number combinations of n object take r at a time is Cn , r nCr n! r!(n r )! where n and r are whole numbers and n r .