advertisement

Thinking Mathematically Counting Methods and Probability 11.4 Fundamentals of Probability Computing Theoretical Probability If an event E has n(E) equally-likely outcomes and its sample space S has n(S) equally-likely outcomes, the theoretical probability of event E denoted by P(E), is num berof outcom esin event E P( E ) num berof outcom esin sam plespace S Example Probability and a Deck of 52 Cards You are dealt one card from a standard 52card deck. Find the probability of being dealt a King. Solution Because there are 52 cards, the total number of possible ways of being dealt a single card is 52. We use 52, the total number of possible outcomes, as the number in the denominator. Because there are 4 kings in the deck, the event of being dealt a king can occur 4 ways. P(king) = 4/52 = 1/13 More Examples Exercise 11.4 #5, 21, 29, 37 A die is rolled, what is the probability of rolling a number less than 3? A fair coin is tossed two times, what is the probability of getting two heads? Arbitrarily selecting a family with three children, what is the probability that the family has exactly two male children? A single die is rolled twice, what is the probability of rolling two numbers such that their sum is 5? Thinking Mathematically Counting Methods and Probability 11.5 Probability with the Fundamental Counting Principle, Permutations, and Combinations Example: Permutations Exercise 11.5 #3 Six stand-up comics, A, B, C, D, E, and F are to perform on a single evening. The order of performance is determined randomly. Find the probability that a. Comic E will perform first b. Comic C will perform fifth and comic B will perform last. c. The comedians will perform in the following order D, E, C, A, B, F. d. Comic A or comic B will perform first. Example: Combinations Exercise 11.5 # 5, #17 A group consists of four men and five women. Three people are selected to attend a conference. a. b. c. In how many ways can three people be selected from this group of nine? In how many ways can three women be selected from the five women? Find the probability that the selected group will consist of all women. If you are dealt 4 cards from a shuffled deck of 52, find that probability of getting two queens and two kings. Thinking Mathematically Counting Methods and Probability 11.6 Events Involving Not and Or The Probability of an Event Not Occurring The probability that an event E will not occur is equal to 1 minus the probability that it will occur. P(not E) = 1 - P(E) Exercise 11.6 #1 If you are dealt one card from a deck of 52, what is the probability that the card is not an ace. Mutually Exclusive Events If it is impossible for events A and B to occur simultaneously, the events are said to be mutually exclusive. If A and B are mutually exclusive events, then P(A or B) = P(A) + P(B). Exercise 11.6 #17 One card is randomly selected from a deck of 52. What is the probability that the card is either a 2 or a 3? Or Probabilities with Events That Are Not Mutually Exclusive If A and B are not mutually exclusive events, then P(A or B) = P(A) + P(B) - P(A and B) Exercise 11.6 #33 You spin a spinner which has 8 equally likely outcomes, numbered 1 through 8. What is the probability that your spin will be either an even number or a number greater than 5? Thinking Mathematically Counting Methods and Probability 11.7 Events Involving And; Conditional Probability Independent Events Two events are independent events if the occurrence of either of them has no effect on the probability of the other. If A and B are independent events, then P(A and B) = P(A)•P(B) Compare to P(A or B) Example: And with Independent Events Exercises 11.7 #3, 13, 19 • A spinner has 3 red regions, 2 green regions, and 1 yellow regions. Each of the 6 regions are equally likely. If the spinner is spun twice, what is the probability of two successive yellow spins? • A single die is rolled twice. What is the probability of rolling an even number on the first roll and a number greater than 2 the second time. • A card is drawn from a deck of 52. The card is replaced, the deck shuffled and a second card is drawn. What is the probability of drawing a red card each time. Dependent Events Two events are dependent events if the occurrence of one of them has an effect on the probability of the other. And Probabilities with Dependent Events If A and B are dependent events, then P(A and B) = P(A)•P(B given that A has occurred) The conditional probability of B, given A, written P(B|A), is the probability that event B occurs computed on the assumption that event A occurs. Example: And with Dependent Events Exercise 11.7 #43 An ice chest contains six cans of apple juice, eight cans of grape juice, four cans of orange juice, and two cans of mango juice. If you randomly choose three cans in succession, what is the probability of choosing three cans of apple juice? Thinking Mathematically Chapter 11 Counting Methods and Probability