4 1.19 Suppose you throw five dice and all outcomes are equally likely.What is the probability that all dice are the same? (In the game of Yahtzee, this is known as a yahtzee.)What is the probability of getting at least one 4?What is the probability that all the dice are different?
5 1.20 Tori is picking her fall term classes. She needs to fill three time slots, and there are 20 distinct courses to choose from, including probability 101, 102, and 103. She will pick her classes at random so that all outcomes are equally likely.What is the probability that she will get probability 101?What is the probability that she will get probability 101 and probability 102?What is the probability she will get all three probability courses?
6 1.21 Suppose numbers are chosen from , where , sampling without replacement. All outcomes are equally likely. What is the probability that the numbers chosen are in increasing order?
7 1.22 There are 40 pairs of shoes in Bill's closet. They are all mixed up.If 20 shoes are picked, what is the chance that Bill's favorite sneakers will be in the group?If 20 shoes are picked, what is the chance that at most one shoe from each of the 40 pairs will be picked? (Remember, a left shoe is different than a right shoe.)
8 1.23 Many bridge players believe that the most likely distribution of the four suits (spades, hearts, diamonds, and clubs) in a bridge hand is 4-3-3-3 (four cards in one suit, and three cards of the other three).Show that the suit distribution 4-4-3-2 is more likely than 4-3-3-3.In fact, besides the 4-4-3-2 distribution, there are three other patterns of suit distributions that are more likely than 4-3-3-3. Can you find them?
9 1.24 Find the probability that a bridge hand contains a nine-card suit. That is, the number of cards of the longest suit is nine.
10 1.25 A chessboard is an eight-by-eight arrangement of 64 squares. Suppose eight chess pieces are placed on a chessboard at random so that each square can receive at most one piece. What is the probability that there will be exactly one piece in each row and in each column?
11 1.26 Find the probabilities for being dealt the following poker hands. They are arranged in increasing order of probability.Straight flush. (Five cards in a sequence and of the same suit.)Four of a kind. (Four cards of one face value and one other card.)Full house. (Three cards of one face value and two of another face value.)Flush. (Five cards of the same suit. Does not include a straight flush.)Straight. (Five cards in a sequence. Does not include a straight flush. Ace can be high or low.)Three of a kind. (Three cards of one face value. Does not include four of a kind or full house.)Two pair. (Does not include four of a kind or full house.)One pair. (Does not include any of the aforementioned conditions.)
12 1.27 A walk in the positive quadrant of the plane consists of a sequence of moves, each one from a point to either or .Show that the number of walks from the origin to is Suppose a walker starts at the origin and at each discrete unit of time moves either up one unit or to the right one unit each with probability 1/2. If , find the probability that a walk from (0,0) to always stays above the main diagonal.
13 1.28 See Example 1.24 for a description of the Powerball lottery. A $100 prize is won by either (i) matching exactly three of the five balls and the powerball or (ii) matching exactly four of the five balls and not the powerball. Find the probability of winning $100.
14 1.29 Give a combinatorial argument (not an algebraic one) for why
15 1.30 Give a combinatorial proof that(1.8) Hint: How many ways can you choose people from a group of men and women? From Equation 1.8 show that(1.9)
Properties of Probabilities
1 1.31 Suppose , , and Find(neither event occurs).
2 1.32 Suppose , , and Find
3 1.33 Suppose and are mutually exclusive, with and Find the probability thatAt least one of the two events occurs.Both of the events occur.Neither event occurs.Exactly one of the two events occur.
4 1.34 Suppose and . Find .FIGURE 1.5: Venn diagram.
5 1.35 Let , , , be three events. At least one event always occurs. But it never happens that exactly one event occurs. Nor does it ever happen that all three events occur. If and , find .
6 1.36 See the assignment of probabilities to the Venn diagram in Figure 1.5. Find the following:(No events occur).(Exactly one event occurs).(Exactly two events occur).(Exactly three events occur).(At least one event occurs).(At least two events occur).(At most one event occurs).(At most two events occur).
7 1.37 Suppose that probabilities have been assigned to the Venn Diagram in Figure 1.5 as follows: , , and . Find the following:(No events occur).(Exactly two events occur).(At most one event occurs).(At most two events occur).
8 1.38 For three events , , and , the following is known: , , , and the probability of no events occurring is 0.1.Sketch a Venn diagram that matches the information provided about the three events.Find .Find .Find .
9 1.39 Four coins are tossed. Let be the event that the first two coins both come up heads. Let be the event that the number of heads is odd. Assume that all 16 elements of the sample space are equally likely. Describe and find the probabilities of (i) , (ii) , and (iii) .
10 1.40 Two dice are rolled. Let be the maximum number obtained. (Thus, if 1 and 2 are rolled, ; if 5 and 5 are rolled, .) Assume that all 36 elements of the sample space are equally likely. Find the probability function for . That is, find , for .
11 1.41 Judith has a penny, nickel, dime, and quarter in her pocket. So does Kory. They both reach into their pockets and choose a coin. Let be the greater (in cents) of the two.Construct a sample space and describe the events for Assume that coin selections are equally likely. Find the probabilities for each of the aforementioned four events.What is the probability that Judith's coin is worth more than Kory's? (It is not )
12 1.42 A tetrahedral dice is four-sided and labeled with 1, 2, 3, and 4. When rolled it lands on the base of a pyramid and the number rolled is the number on the base. In five rolls, what is the probability of rolling at least one 2?
13 1.43 LetShow that is a probability function. That is, show that the terms are nonnegative and sum to 1.Let be defined such that for Find without summing an infinite series.
14 1.44 The functionis a probability function for some choice of . Find .
15 1.45 Let , , be three events. Find expressions for the events:At least one of the events occurs.Only occurs.At most one of the events occurs.All of the events occur.None of the events occur.
16 1.46 The odds in favor of an event is the ratio of the probability that the event occurs to the probability that it will not occur. For example, the odds that you were born on a Friday, assuming birthdays are equally likely, is 1 to 6, often written or 1 to 6, obtained from (1/7)/(6/7).In Texas Hold'em Poker, the odds of being dealt a pair (two cards of the same denomination) is . What is the chance of not being dealt a pair?For sporting events, bookies usually quote odds as odds against, as opposed to odds in favor. In the Kentucky Derby horse race, our horse Daddy Long Legs was given 9 to 2 odds. What is the chance that Daddy Long Legs wins the race?
17 1.47 An exam had three questions. One-fifth of the students answered the first question correctly; one-fourth answered the second question correctly; and one-third answered the third question correctly. For each pair of questions, one-tenth of the students got that pair correct. No one got all three questions right. Find the probability that a randomly chosen student did not get any of the questions correct.
18 1.48 Suppose , , , For each of the events given next, write the event using set notation in terms of , , and , and compute the corresponding probability.At least one of the three events occur.At most one of the three events occurs.All of the three events