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Chapter 5 Probability

Chapter 5 Probability. 5.4 Conditional Probability and the General Multiplication Rule. Conditional Probability – The notation P(F|E) is read “the probability of event F given event E.” It is the probability that the event F occurs given that the event E has occurred.

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Chapter 5 Probability

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  1. Chapter 5Probability

  2. 5.4 Conditional Probability and the General Multiplication Rule • Conditional Probability – The notation P(F|E) is read “the probability of event F given event E.” It is the probability that the event F occurs given that the event E has occurred. • Example: Suppose that a single die is rolled. What is the probability that the die comes up 3? Now suppose that the die is rolled a second time, but we are told the outcome will be an odd number. What is the probability that the die comes up 3?

  3. Conditional Probability Rule • P(F|E) = P(E and F)/P(E) • Example: What is the probability of drawing a King given it is a Heart? • E = Heart • F = King • How many cards are the King of Hearts? • How many cards are Hearts? • P(F|E) = 1/13

  4. Example • In 2005, 12.64% of all births were preterm. Also in 2005, 0.22% of all births resulted in a preterm baby that weighed 8 pounds, 13 ounces or more. What is the probability that a randomly selected baby weighs 8 pounds, 13 ounces or more, given that the baby was preterm? Is this unusual? • E = Preterm baby • F = a baby that weighs at least 8 pounds, 13 ounces • What is the percentage of babies who were preterm and weighed at least 8 pounds, 13 ounces? • What is the percentage of babies who were preterm? • P(F|E) = 0.0022/0.1264 = 0.0174

  5. General Multiplication Rule • The probability that two events E and F both occurs is P(E and F) = P(E) * P(F|E) • Example: Two cards are drawn from a deck of cards without replacement. Find the probability that both of the cards are Hearts? • E = first card is a Heart • F = second card is a Heart • Are the events independent? • P(E) = 13/52 • P(F|E) = 12/51 • P(E and F) = 13/52 * 12/51 = 0.059

  6. Example • The probability that a driver who is speeding gets pulled over is 0.8. The probability that a driver gets a ticket, given that he or she is pulled over, is 0.9. What is the probability that a randomly selected driver who is speeding gets pulled over and gets a ticket? • E = Driver who is speeding gets pulled over • F = Driver gets a ticket • P(E) = 0.8 • P(F|E) = 0.9 • P(F|E) = 0.8 * 0.9 = 0.72

  7. 5.5 Solve Counting Problems Using the Multiplication Rule • Multiplication Rule of Counting – If a task consists of a sequence of choices in which there are p selections for the first choice, q selections for the second choice, r selections for the third choice, and so on, then the task of making these selections can be done in p*q*r*… different ways.

  8. Example • The International Airline Transportation Association (IATA) assigns three-letter codes to represent airport locations. For example, the code for Columbia International Airport is CAE. How many different airport codes are possible? • How many letters can be picked for the first letter? • How many letters can be picked for the second letter? • How many letters can be picked for the third letter? • 26 * 26 * 26 = 263 = 17,576

  9. Example • Three members from a 14 member committee are to be randomly selected to serve as chair, vice-chair, and secretary. The first person selected is the chair, the second is the vice-chair, and the third is the secretary. How many different committee structures are possible? • How many ways can the chair be picked? • How many ways can the vice-chair be picked? • How many ways can the secretary be picked? • 14 * 13 * 12 = 2,184

  10. Factorials • If n >= 0 is an integer, the factorial symbol, n!, is defined as follows: n! = 1*2*3*…*(n-1)*n 0! = 1 1! = 1 • Examples: 3! = 1 * 2 * 3 = 6 5! = 1 * 2 * 3 * 4 * 5 = 120

  11. Example • You have just been hired as a book representative for Pearson Education. On your first day, you must travel to seven schools to introduce yourself. How many different routes are possible? • How many schools can you visit first? • How many schools can you visit second? • 7 * 6 * 5 * 4 * 3 * 2 * 1 = 7! = 5,040

  12. Permutations • A permutation is an ordered arrangement in which r objects are chosen from n distinct (different) objects and repetition is not allowed. The symbol nPr represents the number of permutations of r objects selected from n objects. • Number of Permutations of n Distinct Objects Taken r at a Time • The number of arrangements of r objects chosen from n objects, in which • The n objects are distinct • Repetition of objects is not allowed, and • Order is important • is given by the formula nPr = n!/(n – r)!

  13. Example • Horse racing has a bet called the trifecta. The trifecta is picking the horse that wins (first place), places (second place), and shows (third place) in that order. The Kentucky Derby last year had 20 horses. How many different ways can the horses finish win, place, and show? • 20P3 = 6,840 • The payout for the trifecta at the Kentucky Derby was $3952.40 for a $2 bet. What is the probability you get the correct trifecta on one $2 bet? • 1/6,840 = 1.462e-4 = 0.0001462

  14. Combinations • A combination is a collection, without regard to order, of n distinct objects without repetition. The symbol nCr represents the number of combinations of n distinct objects taken r at a time. • Number of Combinations of n Distinct Objects Taken r at a Time • The number of different arrangements of n objects using r <= n of them, in which • the n objects are distinct, • repetition of objects is not allowed, and • order is not important • is given by the formula nCr = n!/r!(n-r)!

  15. Example • How many different simple random samples of size 4 can be obtained from a population whose size is 20? • Does the order matter? • 20C4 = 4,845 different samples of size 4

  16. Permutations with Nondistinct Items • The number of permutations of n objects of which n1 are of one kind, n2 are of a second kind, …, and nk are of a kth kind is given by n!/(n1! * n2! * … * nk!) where n = n1 + n2 + … + nk

  17. Example • How many different vertical arrangements are there of 10 flags if 5 are white, 3 are blue, and 2 are red? • What is n1? • What is n2? • What is n3? • n = 5 + 3 + 2 = 10 • 10!/(5! * 3! * 2!) = 3,628,800/1440 = 2520

  18. Multiple Techniques • Powerball Lottery consists of white balls numbered 1 to 59 and then red powerballs that are numbered 1 to 36. A customer must pick 5 white balls and 1 red powerball. The order of the white balls do not matter but the red ball must be matched exactly. How many different numbers are there in the lottery? • Does the order matter for the white balls? • 59C5 = 5,006,386 • The powerball comes next. We use the multiplication rule of counting. • 5,006,386 * 36 = 180,229,896 • What is the probability that you will win the Powerball Lottery? • 1/180,229,896 = 5.548e-9 = 0.000000005548 = 0.00000055448%

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