Probability Topic 5: Probabilities Using Counting Methods

I can solve a contextual problem that involves probability and permutations. I can solve a contextual problem that involves probability and combinations.

Explore… As a volunteer activity, 10 students want to put on a talent show at a retirement home. To organize the show, 3 of these students will be chosen at random to form a committee. Victoria really wants to be on this committee, since her grandmother lives at the home. Each student’s name will be written on a slip of paper and placed in a hat. Then 3 names will be drawn. 1. Is Victoria’s name just as likely to be drawn as any other name? Explain. Try the Explore on your own first. Then look at the solutions on the next slides. Yes, since each person has 1 name slip in the hat.

Explore… 2.Does the order in which the names are drawn matter? Explain. 3. In how many different ways can 3 names be drawn from a hat with 10 names. Explain. 4. In how many different ways can Victoria’s name be drawn with 2 other names? Explain. 5. What is the probability that Victoria’s name will be drawn? Explain. No, since all names drawn become equal members on a committee. 10 C 3 = 120 ( 1 C 1 )( 9 C 2 ) = 36

Explore… 6. Did you assume that this problem involved a situation with or without replacement? Explain. 7. Suppose that only 8 students, including Victoria, volunteered for the talent show. Would her name be more likely or less likely to be drawn from the hat? Justify your decision. Without replacement. They wouldn’t put the name back in, since they don’t want to draw the same name twice. More likely to be drawn.

Information In the Permutations and Combinations unit, we learned about three counting techniques: the fundamental counting principle (FCP), permutations, and combinations

Information Probability can be calculated using any of these techniques. The probability of an event E is: S is the number of successes or favourable outcomes to event E. T is the total number of possibilities, with no restrictions. The three counting methods above can be used to find both S and T.

Example 1 To advertise his radio show, Beau decides to hold a contest in the mall. He spells out the word ALBERTA using letter tiles, then turns them over and mixes them up. Beau asks a shopper to arrange the letters in a row and then turn them over. If the tiles spell ALBERTA, the shopper will win a new car. a) What is the probability that the shopper will win the car? Solving a probability problem S - the number of favourable outcomes. N - the total number of outcomes. First we consider the number of ways the letters can be arrangements that spell Alberta. There is exactly 1.

Example 1 b) What is the probability that the shopper will not win the car? Solving a probability problem Since there are 2520 ways to arrange the letters, and only one of these spells Alberta, there are 2519 that do not spell Alberta. N - the total number of outcomes.

Example 2 In the lottery game The Plus a six-digit number is printed on a ticket. Any digit of the six-digit number may be 0 to 9. To win a player needs to have the digits in the correct order. Find the probability that a player’s six- digit number: a) matches all 6 digits of the winning six-digit number. Solving a probability problem S - the number of favourable N - the total number __ __ __ W W W First we consider the number of ways that fit the criteria. All 6 digits must be wining digits. Then we consider the total number of ways that the numbers can be arranged. __ __ __ W W W

Example 2 b) matches exactly the last 5 digits of the six-digit winning number. Solving a probability problem S - the number of favourable N - the total number __ __ __ L W W W W W First we consider the number of ways that fit the criteria. The last 5 digits must be winning numbers and the first digit must be a losing digit. We already know that there are total outcomes.

Example 3 In the lottery game Lotto 649 a player selects 6 numbers between 1 and 49, in any order. What is the probability that a player’s six numbers: a) all match the 6 winning numbers. b) match exactly five of the 6 winning numbers. Solving a probability problem Consider the number of ways that fit the criteria. There are 6 winning numbers and we are selecting all 6 of them. Then we consider the total number of ways that 6 numbers can be selected from 49. Consider the ways that fit the criteria. There are 6 winning numbers and we are selecting 5 of them and there are 43 losing numbers and we are selecting 1 of them. Then we consider the total number of ways that 6 numbers can be selected from 49.

Example 4 There are many different bets that a purchaser can make at a horse racetrack. a) The Quinella is a betting transaction in which the purchaser attempts to select the first two horses to finish, in either order of the finish. In a field of 8 horses, what is the probability that the purchaser wins the Quinella by randomly selecting the horses? Solving a probability problem using counting techniques __ 1st 2 nd First we consider the number of ways that the 2 horses he picks finish in 1 st and 2 nd place (in any order). Then we consider the total number of ways that the 1 st two places are awarded. __ 1st 2 nd

Example 4 b) The Exactor is a betting transaction in which the purchaser attempts to select the first two horses to finish, in the exact order of the finish. In a field of 8 horses, what is the probability that the purchaser wins the Exactor by randomly selecting the horses? Solving a probability problem using counting techniques __ 1st 2 nd First we consider the number of ways that the 2 horses he picks finish in 1 st and 2 nd place (in exact order). Then we consider the total number of ways that the 1 st two places are awarded. __ 1st 2 nd

Example 5 Ewan is opening a new bank account at the local bank branch near his home. A randomly selected 4-digit PIN for his new bank card will be mailed to him within the week. All digits from 0 to 9 can be used and all digits may be used more than once. All PINs are equally likely to be generated. a) What is the probability that the PIN will start with a 7? Solving a probability problem using counting techniques First we consider the number of ways that the PIN could start with a 7. Then we consider the total number of possible PINs.

Example 5 b) What is the probability that the PIN will be an odd number that start with a 7? Solving a probability problem using counting techniques First we consider the number of ways that the PIN could start with a 7 and be odd (end in a 1,3,5,7, or 9. Then we consider the total number of possible PINs.

Example 5 c) What is the probability that all digits in the PIN will be different? Solving a probability problem using counting techniques First we consider the number of ways that the PIN could be made up of all different digits. Then we consider the total number of possible PINs.

Need to Know In order to solve probability problems with many possible outcomes, the context of each particular problem will determine which of the following counting techniques should be used: FCP, the fundamental counting principle, permutations, arranging items when order matters, and combinations, choosing items when order does not matter. You’re ready! Try the homework from this section.