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Magazine Problem 30
What is the probability that a player of a lottery wins the prize offered for correctly choosing five (but not six) numbers out of six integers chosen at random from the integers between 1 and \(40,\) inclusive?
Problem 34
Find each of the following probabilities when n independent Bernoulli trials are carried out with probability of success p. a) the probability of no successes b) the probability of at least one success c) the probability of at most one success d) the probability of at least two successes
Problem 35
In roulette, a wheel with 38 numbers is spun. Of these, 18 are red, and 18 are black. The other two numbers, which are neither black nor red, are 0 and \(00 .\) The probability that when the wheel is spun it lands on any particular number is 1\(/ 38\) . a) What is the probability that the wheel lands on a red number? b) What is the probability that the wheel lands on a black number twice in a row? c) What is the probability that the wheel lands on 0 or 00\(?\) do? d) What is the probability that in five spins the wheel never lands on either 0 or 00\(?\) e) What is the probability that the wheel lands on one of the first six integers on one spin, but does not land on any of them on the next spin?
Problem 36
Use Chebyshev's inequality to find an upper bound on the probability that the number of tails that come up when a biased coin with probability of heads equal to 0.6 is tossed \(n\) times deviates from the mean by more than \(\sqrt{n}\) .
Problem 37
Which is more likely: rolling a total of 9 when two dice are rolled or rolling a total of 9 when three dice are rolled?
Problem 38
A pair of dice is rolled in a remote location and when you ask an honest observer whether at least one die came up six, this honest observer answers in the affirmative. a) What is the probability that the sum of the numbers that came up on the two dice is seven, given the information provided by the honest observer? b) Suppose that the honest observer tells us that at least one die came up five. What is the probability the sum of the numbers that came up on the dice is seven, given this information?
Problem 40
A player in the Powerball lottery picks five different integers between 1 and 69 , inclusive, and a sixth integer between 1 and \(26,\) which may duplicate one of the earlier five integers. The player wins the jackpot if all six numbers match the numbers drawn. a) What is the probability that a player wins the jackpot? b) What is the probability that a player wins \(\$ 1,000,000\) , which is the prize for matching the first five numbers, but not the sixth number, drawn? c) What is the probability that a player wins \(\$ 100\) by matching exactly three of the first five and the sixth numbers drawn, or four of the first five numbers, but not the sixth number, drawn? d) What is the probability that a player wins a prize of \(\$ 4,\) which is the prize when the player matches the sixth number, and either one or none of the first five numbers drawn?
Problem 42
Two events \(E_{1}\) and \(E_{2}\) are called independent if \(p\left(E_{1} \cap E_{2}\right)=p\left(E_{1}\right) p\left(E_{2}\right) .\) For each of the following pairs of events, which are subsets of the set of all possible outcomes when a coin is tossed three times, determine whether or not they are independent. a) \(E_{1} :\) tails comes up with the coin is tossed the first time; \(E_{2} :\) heads comes up when the coin is tossed the second time. b) \(E_{1} :\) the first coin comes up tails; \(E_{2} :\) two, and not three, heads come up in a row. c) \(E_{1} :\) the second coin comes up tails; \(E_{2} :\) two, and not three, heads come up in a row. (We will study independence of events in more depth in Section \(7.2 . )\)
Problem 45
This problem was posed by the Chevalier de M茅r茅 and was solved by Blaise Pascal and Pierre de Fermat. a) Find the probability of rolling at least one six when a fair die is rolled four times. b) Find the probability that a double six comes up at least once when a pair of dice is rolled 24 times. Answer the query the Chevalier de M茅r茅 made to Pascal asking whether this probability was greater than 1\(/ 2\) . c) Is it more likely that a six comes up at least once when a fair die is rolled four times or that a double six comes up at least once when a pair of dice is rolled 24 times?
Problem 47
When \(m\) balls are distributed into \(n\) bins uniformly at random, what is the probability that the first bin remains empty?
