Only correct answers please!!
20. Suppose you have an urn containing 7 red and 3 blue balls. You draw three balls at random. On each draw, if the ball is red you set it aside and if the ball is blue you put it Practice Exam 1: All Questions, Spring 2014 4 back in the urn. What is the probability that the third draw is blue? (If you get a blue ball it counts as a draw even though you put it back in the urn.) 21. Some games, like tennis or ping pong, reach a state called deuce. This means that the score is tied and a player wins the game when they get two points ahead of the other player. Suppose the probability that you win a point is p and this is true independently for all points. If the game is at deuce what is the probability you win the game? This is a tricky problem, but amusing if you like puzzles.marble is drawn. ) List a sample space containing four outcomes. List a sample space with sixteen outcomes. Write the probability of each of the four outcomes in (a) What are the probabilities of the outcomes in (b)? What is the probability the colors of the two marbles match? What is the probability the same marble is drawn twice? 2. We are playing with a short deck, as shown at right. AV A+ A& AC Let "H" be the event the card drawn is a heart. 2V 2+ 24 20 Let "D" be the event the card drawn is a diamond. 37 3+ Let "A" be the event the card is an ace. 44 a. P(H)= P(D) = P(A) = b . P(H or D) = C. P(H or A) = d. P(H and D) = P(H and A) = Are H and D independent events? Are H and A independent events? 3. If three cards are drawn from the deck in #2, one at a time, what is the probability that a. the 1" card is the ace of hearts, the 2" is the 2 of diamonds, and the 3" is the 3 of clubs? b. all three cards are aces? 4. An airplane is built to be able to fly on one engine. If the plane's two engines operate independently, and each has a 1% chance of failing in any given four-hour flight, what is the chance the plane will fail to complete a four-hour flight to Oklahoma due to engine failure? 5. A pair of fair, standard dice are rolled. What is the probability the sum of the dice is 5? 5. Fifty marbles are to be drawn from the jar in problem #1 with replacement. If the first four marbles drawn are red, what is the probability the next marble drawn will not be red? 7 . A probability experiment has four possible outcomes: e,, ez, es, e.. The outcome e, is four times as likely as each of the three remaining outcomes. Find the probability of e,. B. What are the odds in favor of rolling a sum of seven in one roll of a pair of fair standard dice? 9. If P(A) = /, and P(B)= 1, and P(B|A) = 1/3, find: a. P(A and B) b. P(A or B) C. P(A| B) 10. The deck of sixteen cards shown in #2 is thoroughly shuffled. Three cards are drawn from the top of the deck, one at a time. What is the probability the third card is an ace? (Hint: There is areally simple, direct solution.) 11 . "The Birthday Problem" (famous) In a roomful of 30 people, what is the probability that at east two people have the same birthday? Assume birthdays are uniformly distributed and there is no leap year complication. (Hint: what is the probablyty that they all have different birth days?) 12. A 1-inch-diameter coin is thrown on a table covered with a grid of lines two inches apart. What is the probability the coin lands in a square without touching any of the lines of the grid? (Hat: in order that the coin not touch any of the grid Ines, where must the center of the coin be?)7 Maurice works at home. At 2 pm he decides to take a break to buy a copy of the Chronicle newspaper. There are three nearby newsagents: Arif, Bob and Carol. However by 2 pm they may have sold all their Chronicles and so have none available. The independent probabilities that they have a Chronicle available at 2 pm are: Arif 0.4 Bob 0.7 Carol 0.25 (a) State the probability that Bob does not have a Chronicle available at 2 pm. (1 mark) (b) Find the probability that none of the three newsagents has a Chronicle available at 2 pm. (3 marks) (c) (i) Find the probability that Bob does not have a Chronicle available at 2 pm but Arif does. (ii) Find the probability that Carol does not have a Chronicle available at 2 pm but Arif does. (3 marks) (d) Maurice decides to visit the newsagents in turn until he obtains a Chronicle or until he has visited all three. He tosses a coin. If it lands heads he will visit the three newsagents in the order Bob, Arif, Carol. If it lands tails he will visit them in the order Carol, Arif, Bob. Find the probability that he will obtain a Chronicle from Arif. (3 marks) 7 Following a flood, 120 tins were recovered from Dharmesh's corner shop. Unfortunately the water had washed off all the labels. Of the tins. 50 contained pet food. 20 contained peas. 35 contained beans and the rest contained soup. (a) Dharmesh selects a tin at random. Find the probability that it: (i) contains soup: (I mark) (ii) does not contain pet food. (1 mark) (b) Dharmesh selects two tins at random (without replacement). Find the probability that: (i) both contain peas; (2 marks) (ii) one contains pet food and the other contains peas. (3 marks) (c) Dharmesh selects three tins at random (without replacement). Find the probability that one contains pet food, one contains peas and one contains beans. (3 marks) (d) Find the probability that Dharmesh will have to open more than two tins before she finds one which does not contain pet food. (3 marks)7 Maurice works at home. At 2 pm he decides to take a break to buy a copy of the Chronicle newspaper. There are three nearby newsagents: Arif. Bob and Carol. However by 2 pm they may have sold all their Chronicles and so have none available. The independent probabilities that they have a Chronicle available at 2 pm are: Arif 0.4 Bob 0.7 Carol 0.25 (a) State the probability that Bob does not have a Chronicle available at 2 pm. (I mark) (b) Find the probability that none of the three newsagents has a Chronicle available at 2 pm. (3 marks) (c) (i) Find the probability that Bob does not have a Chronicle available at 2 pm but Arif does. (ii) Find the probability that Carol does not have a Chronicle available at 2 pm but Arif does. (3 marks) (d) Maurice decides to visit the newsagents in turn until he obtains a Chronicle or until he has visited all three. He tosses a coin. If it lands heads he will visit the three newsagents in the order Bob, Arif, Carol. If it lands tails he will visit them in the order Carol, Arif, Bob. Find the probability that he will obtain a Chronicle from Arif. (3 marks) 7 Following a flood, 120 tins were recovered from Dharmesh's corner shop. Unfortunately the water had washed off all the labels. Of the tins. 50 contained pet food. 20 contained peas. 35 contained beans and the rest contained soup. (a) Dharmesh selects a tin at random. Find the probability that it: (i) contains soup: (I mark) (ii) does not contain pet food. (1 mark ) Dharmesh selects two tins at random (without replacement). Find the probability that: (i) both contain peas: (2 marks) (ii) one contains pet food and the other contains peas. (3 marks) (c) Dharmesh selects three tins at random (without replacement). Find the probability that one contains pet food, one contains peas and one contains beans. (3 marks) (d) Find the probability that Dharmesh will have to open more than two tins before she finds one which does not contain pet food
