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Name ___________________________________ May 4, 2015 Math 2b/3a Probability Test Review Probability Review Part 1: Counting, Pascal’s Triangle, and the Binomial Theorem Review of Counting Question Types: Type 1: You are asked to count the number of distinct ways of arranging a group of objects. a. How many different ways are there of arranging a group of different things in a row/line/list? Example: Mrs. Runge is planting flowers in a window box outside her kitchen window. She plants 8 flowers, each of which is a different color. How many different ways are there of arranging the flowers? b. How many different ways are there of arranging a group of things in a row/line/list when some are repeated? c. How many different ways are there of arranging a group of different things in a circle? Example: Mrs. Runge is planting flowers in a window box outside her kitchen window. She plants 8 flowers: 3 red, 2 yellow, and 3 white. How many distinct ways are there of arranging the flowers? Example: Mrs. Runge is planting a circular garden. She plants 8 flowers around the edge of the garden, each of which are different colors. How many distinct ways are there of arranging the flowers? 8! = 560 3!2!3! 8! = 5040 8 d. How many different ways are there of arranging a group of objects in a circle when some are repeated? Example: Mrs. Runge is planting a circular garden. She plants 8 flowers around the edge of the garden: 3 red, 2 yellow, and 3 white. How many distinct ways are there of arranging the flowers? 8! = 70 3!2!3!8 Name ___________________________________ May 4, 2015 Math 2b/3a Probability Test Review Type 2: You are asked to count the number of ways of picking a few things out of a larger group. a. How many ways are there of picking a few things out of a group, when the order they are selected matters? b. How many ways are there of picking a few things out of a group, when the order they are selected does not matter? Example: There are 10 students running for class office. The student who gets the most votes becomes president, second most becomes vice-present, third becomes treasurer and fourth becomes secretary. How many different groups of class officers are there? Example: There are 10 students running for class office. The four students who get the most votes are elected as class officers, and they each share the different responsibilities. How many different groups of class officers are there? 10P4 = 5040 10C4 = 10! = 210 4!6! 3. Write down the first 7 rows of Pascal’s Triangle. 4. Find each of the following combination numbers using Pascal’s Triangle. Remember this is the weird notation from the book for 6C2 … æ ö æ ö æ ö a. ç 6 ÷ b. ç 5 ÷ c. ç 3 ÷ è 2 ø è 0 ø è 1 ø 5. Expand the following using the Binomial Theorem. Completely simplify your answer. (x + 2)6 6. Find the number of different ways of rearranging the letters in the following names. a. PAUL b. LENNA c. ACADIA d. NIKKI 7. There are 28 plants in the window box outside of Ranc’s. There are 10 with purple flowers, 10 with yellow flowers, and 8 with orange flowers. How many different ways of arranging the plants are there? For this problem, you do not need to multiply/divide to get a single number as your answer. 8. In the courtyard of Mrs. Runge’s apartment building, there is a circular patch of grass with tulips planted along the edge. There are 24 red tulips and 12 yellow tulips. Find the number of different ways the tulips could be arranged. For this problem, you do not need to multiply/divide to get a single number as your answer. Name ___________________________________ May 4, 2015 Math 2b/3a Probability Test Review 9. a. There are 6 Lexpress busses in line in the Center. There is one of each number bus. b. Now suppose there are three #1 busses, one #2 bus, one #3 bus, and one #4 bus. How many different ways could the busses line up? 10. a. Suppose for next year there are 5 sections of Math 2B/3A, and there are 20 math teachers. How many different ways are there of just choosing a teaching team of 5 different teachers? b. Now suppose instead of just choosing a teaching team, the department head needs to also make scheduling assignments. He first chooses a teacher to teach an A block section, then he chooses a second teacher to teach a B block section, up to a fifth teacher to teach an E block section. How many different ways of making teaching assignments are there? 11. You want to make a bouquet of five different types of flowers from the 12 flowers available. How many bouquets could you make? Name ___________________________________ May 4, 2015 Math 2b/3a Probability Test Review Probability Review Part 2: Mutually Exclusive, Independent, and Conditional Probabilities & Expected Value Two events A and B are mutually exclusive if they do not share any outcomes in the sample space. Example: Rolling a 6 and rolling a 3 when you toss a die one time. Non-Example: getting a heads and getting a tails when you flip a coin two times. Two events A and B are independent if the result from one event has no effect on the other. If A and B are independent, then P(A and B) = P(A) * P(B). Example: rolling a 6 and then rolling a 3. Non-Example: being female and being a mother Important Notes: If two events A and B are mutually exclusive, then P(A or B) = P(A) + P (B). If two events A and B are independent, then P(A and B) = P(A) * P (B).__. The probability of an event A not happening is P(not A) = 1 – P(A). If two events A and B are not mutually exclusive, then P(A and B) = P(A) + P(B) – P(A and B) Expected Value finds what one should expect to happen on average (after many trials) in a given situation. You must know the outcomes & their values, the probabilities for each outcome in order to find the expected value. A value greater than 0 is favorable to the person whose perspective is represented in the outcome/value column. If the value is negative, then it is unfavorable and a value of 0 means there is no net gain or loss and this is considered to be fair. Name ___________________________________ May 4, 2015 Math 2b/3a Probability Test Review Practice Problems: 12. A bag contains 7 blue marbles, 5 red marbles, and 2 white marbles. You select two marbles without replacement. a. Find the probability that the two marbles are the same colors. Hint: There are three different ways this can happen. b. Find the probability that the two marbles are different colors. Hint: This should be pretty quick if you use what you found in part a. 13. Suppose you randomly select an LHS student. Use the following events and their probabilities to find the probabilities listed below, or explain why you can’t. S: student is a sophomore, P(S) = 0.23 F: student is a freshman, P(F) = 0.27 B: student has blue, P(B) = 0.17 E: student is taking Earth Science, P(E) = 0.28 M: student’s favorite subject is math, P(M) = 0.90 a. b. c. d. e. f. P(F and M) P(F or S) P(B and E) P(M|S) P(S|F) P(M or E) 14. You play a carnival game where you roll a number cube and flip a coin. The coin has the number 2 on 2 on one side and 6 on the other side. Your score is the sum of the values that appear on the number cube and the coin flip. a. Write out the sample space for this experiment. b. You win a stuffed animal if you score 11 points or more. Find the probability of winning. 15. Consider the number of students who are late to a class the first block of the day. Use the table to the right to find the expected number of students late to class. Explain what this expected value means. Number of students late Probability 0 1 2 3 4 5 6 7 8 9 0.01 0.02 0.03 0.04 0.10 0.20 0.35 0.15 0.07 0.03 16. If a voter is chosen at random from a large group of registered voters, the probability that the voter is a registered Democrat is 0.46 and the probability that the voter is a registered Republican is 0.45. (A voter is not allowed to be registered in both parties.) a. What is the probability that a randomly chosen voter is not a Democrat? b. What is the probability that a randomly chosen voter is a Democrat or a Republican? c. Voters that are neither Democrat nor Republican are called Independent. What is the probability that a randomly chosen voter is an Independent? Name ___________________________________ May 4, 2015 Math 2b/3a Probability Test Review 17. On any given day in May, suppose you wear jeans to school with probability 0.6 and you wear shorts to school with probability 0.3. You wear something else, like sweatpants, with probability 0.1. Let A be the event that you wear jeans, let B be the event that you wear shorts, and let C be the event that you wear something else. Find each of the following probabilities. a. b. c. d. e. f. g. P(A or B) P(A and C) P(A|B) P(not B) P(neither A nor B) P(A or B or C) P(A and B and C) 18. Here are the actual probabilities for the problem about taking 5 cell phones and returning them at random. The table at the right shows the probabilities for how many people get their own phones back. a. Why do all the probabilities have 120 as their denominator? Hint: The number 120 comes from a counting problem. number of people who get back their own phone n=0 n=1 n=2 n=3 n=4 n=5 probability 44 120 45 120 20 120 10 120 0 1 120 b. On average, how many people will get their own phones back? In other words, calculate the expected value for the number of people who get back their own phone. 19. In a drawer there are 7 green tennis balls (G) and 3 pink tennis balls (P). Suppose that one ball is taken from the drawer, and then (without replacing the first one) a second ball is taken. a. Find all the outcomes for this experiment. b. What is the probability that the first tennis ball is pink and the second tennis ball is pink? c. What is the probability that both tennis balls are green? d. What is the probability that the two tennis balls have different colors? e. You make a deal with your sister that if you can pull two pink tennis balls out of the drawer, she has to do your chores for 7 days, if one of the two is pink, she does your chores for 3 days and if both are green you do her chores for 5 days. Who made the better deal (Hint: what is the expected value)?