MAT 155. Key Concept. Notation. Fundamental Counting. February 09, S4.7_3 Counting. Chapter 4 Probability

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1 MAT 155 Dr. Claude Moore Cape Fear Community College Chapter 4 Probability 4 1 Review and Preview 4 2 Basic Concepts of Probability 4 3 Addition Rule 4 4 Multiplication Rule: Basics 4 7 Counting Key Concept In many probability problems, the big obstacle is finding the total number of outcomes, and this section presents several methods for finding such numbers without directly listing and counting the possibilities. Very good! Video tutorials for TI calculator in statistics Community College of Baltimore County TI Calculator tutorials Fundamental Counting For a sequence of two events in which the first event can occur m ways and the second event can occur n ways, the events together can occur a total of m n ways. Notation The factorial symbol! denotes the product of decreasing positive whole numbers. For example, By special definition, 1! = 1 and 0! = 1. 1

Factorial Rule A collection of n different items can be arranged in order n! different ways.

) Permutations Rule (when items are all different) There are n different items available. (This rule does not apply if some of the items are identical to others.

) If the preceding requirements are satisfied, the number of permutations (or sequences) of r items selected from n available items (without replacement) is Permutations Rule (when some items are

2 Factorial Rule A collection of n different items can be arranged in order n! different ways. (This factorial rule reflects the fact that the first item may be selected in n different ways, the second item may be selected in n 1 ways, and so on.) Permutations Rule (when items are all different) There are n different items available. (This rule does not apply if some of the items are identical to others.) We select r of the n items (without replacement). We consider rearrangements of the same items to be different sequences. (The permutation of ABC is different from CBA and is counted separately.) If the preceding requirements are satisfied, the number of permutations (or sequences) of r items selected from n available items (without replacement) is Permutations Rule (when some items are identical to others) There are n items available, and some items are identical to others. We select all of the n items (without replacement). We consider rearrangements of distinct items to be different sequences. Combinations Rule There are n different items available. We select r of the n items (without replacement). We consider rearrangements of the same items to be the same. (The combination of ABC is the same as CBA.) If the preceding requirements are satisfied, the number of combinations of r items selected from n different items is If the preceding requirements are satisfied, and if there are n 1 alike, n 2 alike,... n k alike, the number of permutations (or sequences) of all items selected without replacement is 2

3 Permutations versus Combinations When different orderings of the same items are to be counted separately, we have a permutation problem, but when different orderings are not to be counted separately, we have a combination problem. Recap In this section we have discussed: The fundamental counting rule. The factorial rule. The permutations rule (when items are all different). The permutations rule (when some items are identical to others). The combinations rule. 195/6. Factorial Find the number of different ways that the nine players on a baseball team can line up for the National Anthem by evaluating 9!. 195/8. Card Playing Find the number of different possible five card poker hands by evaluating 52C /10. Scheduling Routes A political strategist must visit state capitols, but she has time to visit only 3 of them. Find the number of different possible routes by evaluating 50P /12. Trifecta Refer to Exercise 3. Find the number of different possible trifecta bets in a race with ten horses by evaluating 10P 3. 3

4 In Exercises 13 16, find the probability of winning the indicated lottery by buying one ticket. In each case, numbers selected are different and order does not matter. Express the result as a fraction. 196/13. Lotto Texas Select the six winning numbers from 1, 2,, 54. In Exercises 13 16, find the probability of winning the indicated lottery by buying one ticket. In each case, numbers selected are different and order does not matter. Express the result as a fraction. 196/14. Florida Lotto Select the six winning numbers from 1, 2,, 53. Total number of possible tickets with 6 different numbers from 1 to 54 is the number combinations of 54 numbers taken 6 at a time. This represents the denominator. So, the probability is P(Win) = 1/25,827,165. P(Win) = 1 / Total possible outcomes Total possible outcomes: 53 C 6 = 22,957,480 P(Win) = 1 / 22,957, /19. Sampling The Bureau of Fisheries once asked for help in finding the shortest route for getting samples from locations in the Gulf of Mexico. How many routes are possible if samples must be taken at 6 locations from a list of 20 locations? 196/21. Electricity When testing for current in a cable with five color coded wires, the author used a meter to test two wires at a time. How many different tests are required for every possible pairing of two wires? 4

5 197/24. Simple Random Sample In Phase I of a clinical trial with gene therapy used for treating HIV, five subjects were treated (based on data from Medical News Today). If 20 people were eligible for the Phase I treatment and a simple random sample of five is selected, how many different simple random samples are possible? What is the probability of each simple random sample? 197/28. Safe Combination The author owns a safe in which he stores all of his great ideas for the next edition of this book. The safe combination consists of four numbers between 0 and 99. If another author breaks in and tries to steal these ideas, what is the probability that he or she will get the correct combination on the first attempt? Assume that the numbers are randomly selected. Given the number of possibilities, does it seem feasible to try opening the safe by making random guesses for the combination? 198/33. Powerball As of this writing, the Powerball lottery is run in 29 states. Winning the jack pot requires that you select the correct five numbers between 1 and 55 and, in a separate drawing, you must also select the correct single number between 1 and 42. Find the probability of winning the jackpot. 198/34. Mega Millions As of this writing, the Mega Millions lottery is run in 12 states. Winning the jackpot requires that you select the correct five numbers between 1 and 56 and, in a separate drawing, you must also select the correct single number between 1 and 46. Find the probability of winning the jackpot. 5

8.3 Probability with Permutations and Combinations

8.3 Probability with Permutations and Combinations Question 1: How do you find the likelihood of a certain type of license plate? Question 2: How do you find the likelihood of a particular committee? Question