Robert P. Dobrow - Probability

2 1.2 Roll four dice. Consider the probability of getting all fives.

3 1.3 A pizza shop offers three toppings: pineapple, peppers, and pepperoni. A pizza can have 0, 1, 2, or 3 toppings. Consider the probability that a random customer asks for two toppings.

4 1.4 Bored one day, you decide to play the video game Angry Birds until you win. Every time you lose, you start over. Consider the probability that you win in less than 1000 tries.

5 1.5 In Julia's garden, there is a 3% chance that a tomato will be bad. Julia harvests 100 tomatoes and wants to know the probability that at most five tomatoes are bad.

6 1.6 In two dice rolls, let be the outcome of the first die, and the outcome of the second die. Then is the sum of the two dice. Describe the following events in terms of simple outcomes of the random experiment: (Example solution: .)....

7 1.7 A bag contains red and blue balls. You reach into the bag and take balls. Let be the number of red balls you take. Let be the number of blue balls. Express the following events in terms of and , assuming valid values for , , and :You pick no red balls. (Example solution: .)You pick one red and two blue balls.You pick four balls.You pick twice as many red balls as blue balls.You pick at least two red balls.

8 1.8 A couple plans to continue having children until they have a girl or until they have six children, whichever comes first. Describe an appropriate sample space for this random experiment.

1 1.9 A sample space has four elements such that is twice as likely as , which is three times as likely as , which is four times as likely as . Find the probability function.

2 1.10 A sample space has four elements such that is ten times as likely as , which is four times as likely as . Finally, is as likely as and combined. Find the probability function.

3 1.11 A sample space has four elements. For the potential probability functions for the sample space below, state whether they are valid or not. Provide support for your response.....

4 1.12 A random experiment has three possible outcomes , , and , with1.13 What choice(s) of makes this a valid probability model?

5 1.13 Let and be two probability functions on . Define a new function such that Show that is a probability function.

6 1.14 Suppose are probability functions on Let be a sequence of numbers. Under what conditions on the 's willbe a probability function?

7 1.15 Let be a probability function on such that and for Let be a function on defined by For what value(s) of will be a valid probability function?

1 1.16 A club has 10 members including Nasir, Rose, and Devin, and is choosing a president, vice-president, and treasurer. All selections are equally likely.What is the probability that Nasir is selected president?What is the probability that Rose is chosen president and Devin is chosen treasurer?What is the probability that neither Nasir, Rose, or Devin obtain a position?

2 1.17 A fair coin is flipped six times. What is the probability that the first two flips are heads and the last two flips are tails? Use the multiplication principle.

3 1.18 Suppose that license plates can be two, three, four, or five letters long, taken from the alphabets A to Z. All letters are possible, including repeats. A license plate is chosen at random in such a way so that all plates are equally likely.What is the probability that the plate is “A-R-R?”What is the probability that the plate is four letters long?What is the probability that the plate is a palindrome?What is the probability that the plate has at least one “R?”

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 1.24for 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)

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.