Unit 9: Probability Assignments #1: Basic Probability In each of exercises 1 & 2, find the probability that the spinner shown would land on (a) red, (b) yellow, (c) blue. 1. 2. Y B B Y B R Y Y B R 3. Suppose the spinner shown here is spun once, to determine a single-digit number, and we are interested in the event E that the resulting number is odd. Give each of the following: (a) the sample space (b) the number of favorable outcomes (c) the number of unfavorable outcomes (d) the total number of possible outcomes (e) the probability of an odd number (f) the odds in favor of an odd number 4. Lynn Damme s group of preschool children includes eight girls and five boys. If Lynn randomly selects one child to be first in line, with E being the event that the one selected is a girl, give each of the following. (a) the total number of possible outcomes (b) the number of favorable outcomes (c) the number of unfavorable outcomes (d) the probability of event E (e) the odds in favor of event E
5. The spinner of Exercise 3 is spun twice in succession to determine a two-digit number. Give each of the following: (a) the sample space (b) the probability of an odd number (c) the probability of a number with repeated digits (d) the probability of a number greater than 30 (e) the probability of a prime number 6. Two fair coins are tossed (say a dime and a quarter). Give each of the following: (a) the sample space (b) the probability of heads on the dime (c) the probability of heads on the quarter (d) the probability of getting both heads (e) the probability of getting the same outcome on both coins 7. Anne Kelly randomly chooses a single ball from the urn shown. The urn contains 4 blue, 5 yellow, and 2 red. Find the odds in favor of each event: (a) red (b) yellow (c) blue
#2: Not/Or Events 1-6. For the experiment of rolling a single fair die, find the probability of each of the following events: 1. Not less than 2 2. Not prime 3. Odd or less than 5 4. Even or prime 5. Odd or even 6. less than 3 or greater than 4 7-12. For the experiment of drawing a single card from a standard 52-card deck, find (a) the probability, and (b) the odds in favor, of each of the following events: 7. Not an ace 8. King or queen 9. Club or heart 10. Spade or a face card 11. Not a heart, or a 7 12. Neither a heart nor a 7 13-16. For the experiment of rolling an ordinary pair of dice, find the probability that the the sum will be each of the following: 13. 11 or 12 14. Even or a multiple of 3 15. Odd or greater than 9 16. Less than 3 or greater than 9 17. An experiment consists of spinning both spinners shown here and multiplying the resulting numbers together. Find the probability that the resulting product will be even. 4 1 10 8 3 2 9
#3: Conditional Events 1-6. One hundred college seniors attending a career fair at a major northeastern university were categorized according to gender and according to primary career motivation, as summarized here: Gender Primary Career Motivation Money Allowed to be Sense of Giving Total Creative To Society Male 18 21 19 58 Female 14 13 15 42 Total 32 34 34 100 If one of these students is to be selected at random, find the probability that the student selected will satisfy each of the following conditions: 1. female 2. motivated primarily by creativity 3. not motivated primarily by money 4. male and motivated primarily by money 5. male, given that primary motivation is a sense of giving to society 6. motivated primarily by money or creativity, given that the student is female 7-10. Find each of the following probabilities when a single card is drawn from a standard 52-card deck. 7. P(queen face card) 8. P(face card queen) 9. P(red diamond) 10. P(diamond red)
#4: And Events 1-2. For each of the following experiments, determine whether the two given events are independent. 1. A fair coin is tossed twice. The events are head on the first and head on the second. 2. A pair of dice are rolled. The events are even on the first and odd on the second. 3-7. Let two cards be dealt successively, without replacement, from a standard 52-card deck. Find the probability of the following events. 3. spade second, given spade first 4. club second, given diamond first 5. two face cards 6. no face cards 7. the first card is a jack and the second is a face card 8-11. A pet store has seven puppies, including four poodles, two terriers, and one retriever. If Rebecka and Aaron, in that order, each select one puppy at random, with replacement, find the probability of each of the following events: 8. both select a poodle 9. Rebecka selects a retriever, Aaron selects a terrier 10. Rebecka selects a terrier, Aaron selects a retriever 11. both select a retriever 12-17. Suppose two puppies are selected as above, but this time without replacement. Find the probability of each of the following events. 12. both select a poodle 13. Aaron selects a terrier, given Rebecka selects a poodle 14. Aaron selects a retriever, given Rebecka selects a poodle 15. Rebecka selects a retriever 16. Aaron selects a retriever, given Rebecka selects a retriever 17. both select a retriever
#5: Bernoulli s Probability Formula 1-5. If three fair coins are tossed, find the probability of each of the following numbers of heads. 1. 0 2. 3 3. 1 or 2 4. at least 1 5. no more than 1 6-7. Find the probabilities of the following numbers of heads when seven fair coins are tossed. 6. 1 7. 2 8-11. A fair die is rolled three times. A 4 is considered success while all other outcomes are failures. Find the probability of each of the following numbers of successes. 8. 0 9. 1 10. 2 11. 3 12. For n repeated independent trials, with constant probability of success p for all trials, find the probability of exactly x successes if n = 5, p = 1 3, and x = 4.
#6: Expected Value 1. Five fair coins are tossed. Find the expected number of heads. 2-3. A certain game consists of rolling a single fair die and pays off as follows: $3 for a 6, $2 for a 5, $1 for a 4, and no pay off otherwise. 2. Find the expected winnings for this game. c. What is a fair price to pay to play this game? 4-5. Consider a game consisting of rolling a single fair die, with payoffs as follows. If an even number of spots turn up, you receive that many dollars. But if an odd number turns up, you must pay that many dollars. 4. Find the expected net winnings of the game. 5. Is this game fair, or unfair against the player, or unfair in favor of the player? 6. A certain game involved tossing 3 fair coins, and it pays 10 for 3 heads, 5 for 2 heads, and 3 for 1 head. Is 5 a fair price to pay for this game? 7-9. Five thousand raffle tickets are sold. One first prize of $1000, two second prizes of $500 each, and five third prizes of $100 each are to be awarded, with all winners selected randomly. 7. If you purchased one ticket, what are your expected winnings? 8. If you purchased two tickets, what are your expected winnings? 9. If the tickets were sold for $1 each, how much profit goes to the raffle sponsor?
#7: Binomial Expansion Use the binomial theorem to expand the expression. Express your answer in simplest form. 1. 2. 3. 3 ( s 3) 4 ( y 2) (2 x y) 3 4. Find the 5 th term of 6 (3k 1) 5. Find the 4 th term of ( x 2 y) 8 6. Write the 4 th term of 7 (4x 1)