Unformatted text preview: Probability Probability Readings Pages 135140 in your book 135Online text book: http://davidmlane.com/hyperstat/probability.html Probability The likelihood of an event occurring expressed as the ratio of the number of actual occurrences to the number of possible occurrences. Probability Event= basic data Event= point/outcome on a single trial e.g., rolling a 4 on a die The study of probability mostly concerns itself with frequency related probability 1 Probability The probability of picking a red marble is 5 out of 7, or 5/7. The probability of picking a blue marble is 2 out of 7 The probability of picking a red marble or a blue marble is 7 out of 7 Independent: Probability Events we'll consider are: we' when the occurrence or nonoccurrence of one event has no effect on the occurrence or nonoccurrence of the other event. Mutually exclusive: if the occurrence of one event precludes the occurrence of the other event. 5 Red Marbles 2 Blue Marbles Probability Range: Proportion: 0 to 1.00 If the probability is equal to 0.00 then the event is certain not to occur. Basic Laws of Probability The Additive Rule: Given a set of mutually exclusive events, the probability of the occurrence of one event or another is equal to the sum of their separate probabilities. For mutually exclusive events "or" or" Percentages: 0%  100% Probabilities as symbols: p p < 0.05 P (Event A U B) = P(A) + P(B) 2 Basic Laws of Probability The General Addition Rule: Nonmutually Nonexclusive events. IF events are NOT mutually exclusive, then there will be some overlap. If we just simply added the probability to find out the P(A) or P(B), then we would be counting that overlap. Basic Laws of Probability The Multiplicative Rule: The probability of the joint occurrence of two or more independent events is the product of their individual probabilities. For independent events "and" and" P (Event A B) = P(A) * P(B) P ( A B ) = P ( A) + P ( B )  P ( A B) Basic Laws of Probability General Multiplicative Rule: Dependent Events. If the occurrence of one event affects the probability of the other event occurring. If we simply multiple to find the P(A) and P(B), then we overlook this relationship. Basic Laws of Probability P(Heads) = 0.50 P(Heads U Tails) = ? (0.50)+(0.50) = 1.00 P(Heads Tails) = ? P( A B) = P ( A) * P( B  A) When the P(B) is dependent upon A. (0.50*0.50) + (0.50*0.50) = 0.50 =P(Heads Tails) U P(Tails Heads) 3 Basic Counting Rules Factorials n! = n * (n1) * (n2) * (n3)*etc... (n(n(n 3)*etc... n=4: 4 * 3 * 2 * 1 Basic Counting Rules Permutations Total of 5 elements (1, 2, ...5), one sequence of three elements would be (1, 3, 4) n = 10 R =3 Special case: 0! = 1 Permutations An arrangement of objects without repetition where order is important. R = size of each permutation n = size of the set from which elements are permutated ! = factorial N To calculate the number of sequences... sequences... PR = N! ( N  r )! N N! NP = R ( N  r )! PR = 10! (10  3)! N PR = 720 Basic Counting Rules Combinations An arrangement of objects without repetition where order is not important. The difference between a permutation and combination is NOT whether there is repetition or not. There must not be repetition in either. The difference is whether order is important or not. Basic Counting Rules A bag of cookies contains 6 chocolate chip, 5 peanut butter, and 1 oatmeal. Billy selects 2 cookies at random. Find the probability that Billy selected 2 chocolate chip (CC) cookies. N = # of objects (6 CC & 12 Total) R = group size (2) N CR = N CR = N! R !( N  R)! N! R !( N  R )! = C2 15 = = 0.227 66 12 C2 6 4 ...
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 Spring '08
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