Section 11.4: Tree Diagrams, Tables, and Sample Spaces


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1 Section 11.4: Tree Diagrams, Tables, and Sample Spaces Diana Pell Exercise 1. Use a tree diagram to find the sample space for the genders of three children in a family. Exercise 2. (You Try!) A soda machine dispenses both Coke and Pepsi products, in both 12 ounce cans and 20ounce bottles. For each brand, it has a regular cola, diet cola, and lemonlime drink. Use a tree diagram to find the sample space for the experiment of choosing one drink at random from this machine. Exercise 3. A coin is flipped, and then a die is rolled. Use a tree diagram to find the probability of getting heads on the coin and an even number on the die. 1
2 Exercise 4. (You Try!) In order to collect information for a student survey, a researcher classifies students according to eye color (blue, brown, green), gender (male, female), and class rank (freshman, sophomore). A folder for each classification is then made up (e.g., freshman/female/green eyes). Find the sample space for the folders using a tree diagram. If a folder is selected at random, find the probability that a) It includes students with blue eyes. b) It includes students who are female. c) It includes students who are male freshmen. Section 11.5: Probability Using Permutations and Combinations Exercise 5. Stacy has the option of selecting three books to read for a humanities course. The suggested book list consists of 10 biographies and five current events books. She decides to pick the three books at random. Find the probability that all three books will be current events books. 2
3 Exercise 6. (You Try!) There are 12 women and 8 men in a seminar course. If the professor chooses fiveperson groups at random, what is the probability that the first group chosen will consist of all women? Exercise 7. What is the probability of getting 4 aces when drawing 5 cards from a standard deck of 52 cards? Exercise 8. (You Try!) Suppose the deck of cards in the example above has all 32 cards with numbers less than 10 removed, so that only 10s, jacks, queens, kings, and aces remain. Now what is the probability of getting 4 aces when drawing 5 cards? Exercise 9. A combination lock has 40 numbers on it, from zero to 39. Find the probability that if the combination to unlock it consists of three numbers, it will contain the numbers 1, 2, and 3 in some order. Assume that numbers cannot be repeated in the combination. (It s interesting to note that a combination lock should really be called a permutation lock since the order of the numbers is important when you are unlocking the lock.) 3
4 Exercise 10. (You Try!) A different?permutation? lock has letters from A through L on it, and the combination consists of four letters with no repeats. What is the probability that the combination is I, J, K, and L in some order? Exercise 11. A store has six different fitness magazines and three different news magazines. If a customer buys three magazines at random, find the probability that the he ll pick two fitness magazines and one news magazine. Exercise 12. (You Try!) Find the probability that the customer in the example above picks at least two fitness magazines. Exercise 13. The list of potential parolees at a monthly parole hearing consists of eight drug offenders, five violent offenders, and two convicted of property crimes. I d surely like to think that parolees aren t chosen at random, but if this particular board chooses three parolees randomly, find the probability that 4
5 a) All three are drug offenders. b) Two of the three are property offenders. c) All three are violent offenders. d) One of each type of offender is paroled. e) Two are drug offenders and one is a violent offender. 5
6 Section 11.6: Odds and Expectations Converting between Odds and Probabilities If the odds in favor of the event E occurring are a to b, then P (E) = a a + b If P (E) = p, then the odds in favor of E are found by reducing the fraction p to the form a, 1 p b where a and b are integers having no common divisor. Then the odds in favor of E are. a to b Exercise 14. What are the odds of obtaining a three when rolling a die. Exercise 15. The probability of obtaining a sum of eight or more when rolling a pair of dice is. What are the odds of obtaining a sum of eight or more? Exercise 16. Four people are running for class president: Liz, John, Sue, and Tom. probabilities of John, Sue, and Tom winning are.18,.23, and.31, respectively. The (a) What is the probability of Liz winning? (b) What is the probability that a boy wins? 6
7 (c) What is the probability that Tom loses? (d) What are the odds that Sue loses? (e) What are the odds that a girl wins? (f) What are the odds that John wins? Exercise 17. (You Try!) A card is drawn from a standard deck of 52 cards. (a) Find the odds in favor of getting an ace. (b) Find the odds against getting an ace. 7
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