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A Survey of Probability Concepts Chapter Five McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. Descriptive Statistics deals with what happened in the past. Inferential Statistics deals with what may happen in the future. In studying about future events, we begin with the concept of probability. Expressed as a decimal between 0 and 1 Classical Probability (assumes all outcomes are equally likely) Probability of an event = No of possible favorable outcomes Total number of possible outcomes Eg. When you throw a die, what is P(Even) ? Possible even numbers you can get are: 2, 4 & 6 (3 possible favorable outcomes) Total possible outcomes are: 6 (ie. You can get 1,2,3…6) So, P(Even) = 3/6 = 0.5 You can think of card games, lotteries as examples of classical probability. Events are Mutually Exclusive if the occurrence of any one event means that none of the others can occur at the same time. Mutually exclusive: Rolling a 2 precludes rolling a 1, 3, 4, 5, 6 on the same roll. Events are Collectively Exhaustive if at least one of the outcomes must occur when an experiment is conducted. If you throw a die, you must get either 1 or 2 or…6; ie. one of the events must occur. Rules for Computing Probabilities (Memorize, ie. thru understanding!) Complement rule: P(A) = 1 – P(~A) Special rule of addition: P(A or B) = P(A) + P(B) General rule of addition: P(A or B) = P(A) + P(B) – P(A & B) Special rule of multiplication: P(A and B) = P(A)xP(B) General rule of multiplication: P(A and B) = P(A)xP(B|A) or = P(B)xP(A|B) No. of combination from n objects taken r at a time: nCr = n! / [r! (n-r)!] Special Rule of Addition If two events A and B are mutually exclusive, the Probability of A or B equals the sum of their respective probabilities. Checkout problem #14 (a) on page 133 P(A or B) = P(A) + P(B) The Complement rule using a Venn Diagram P(A) + P(~A) = 1; thus, P(A) = 1 – P(~A) Set of all possible outcomes A A subset of outcomes ~A Checkout problem #14 (b) on page 133 ~A is called the complement of A The possible remaining outcomes The General Rule of Addition If A and B are two events that are not mutually exclusive, then P(A or B) is given by the following formula: P(A or B) = P(A) + P(B) - P(A and B) Joint probability What is the probability a tourist visited Disney World OR Busch Gardens? P(Disney OR Busch) = P(Disney) + P(Busch) – P(Disney AND Busch) = .6 Venn diagram + .5 - .3 = .8 The area that is ‘double-counted’ must be subtracted. This is what the formula does. Note: If a tourist is allowed to visit only one park (ie mutually exclusive), then P(Disney and Busch)=0. Then, the general rule becomes the special rule! The Special Rule of Multiplication This rule is written: P(A and B) = P(A) x P(B) Events A and B must be independent. Independence means occurrence of one event has no effect on the probability of the occurrence of the other. Eg. You toss 2 coins (or a coin two times). What is P(2 Heads)? P(2Heads) = P(I coin is head) x P(II coin is also head) Note that the outcome of one coin does not influence the other. So, P(2 Heads) = .5 x .5 = .25 The General Rule of Multiplication Events A and B are not independent. ie. One event has effect on the probability of the occurrence of the other. The rule is written: P(A and B) = P(A)P(B|A) P(A and B) = P(B)P(A|B) or Notes: The probability of event A occurring given that the event B has occurred is written P(A|B). It is called a Conditional probability. If A and B are independent, P(A|B) = P(A). Think about a coin toss expt. Whether you get a H or T is not dependent on the outcome of the prior toss. Question 1: If a student is selected at random, what is the probability the student is a female? P(F) = 400/100 (see Table in prior slide) Question 2: What is the probability, the student is a female accounting major? P(F and A) = 110/1000 (again, see prior Table) Question 3: Given that the student is a female, what is the probability that she is an accounting major? ie Find P(A|F) Major We know from the General rule that Accounting P(F and A) = P(F) x P(A|F) Finance Re-arranging the formula, Marketing P(A|F) = P(F and A) / P(F) Management = [110/1000] / [400/1000] Total = .275 Male 170 Female 110 Total 280 120 100 220 160 70 230 150 120 270 600 400 1000 A Tree Diagram is useful for portraying conditional and joint probabilities. It is particularly useful for analyzing business decisions involving several stages. Based on Self-Review 5-8 Page 140 Convenient Visits Yes No Total Often 60 20 80 Occasio nal 25 35 60 Never 5 50 55 90 105 195 1.00 A Combination is the number of ways to choose r objects from a group of n objects where order does not matter. n! nCr r! (n r )! where, n! = 1x2x3x ..... (n-2)x(n-1)xn You want to select a committee of two out of three people. How many ways are possible? 3C2 = 3! / [2! (3-2)!] = 3 ways Say, these three people are: Mr.A, Mrs.B and Ms.C. The three possible committee combinations are: A,B or A,C or B,C