Chapter 7: Discrete Probability

Question: Let E and F be the events that a family of$\mathit{n}$children has children of both sexes and has at most one boy, respectively. Are E and F Independent if

role="math" localid="1668598698436" $$\begin{gathered} \mathit{a}\mathbf{)}\mathbf{}\mathit{n}\mathbf{=}\mathbf{2} \\ \mathit{b}\mathbf{)}\mathit{n}\mathbf{=}\mathbf{4}\mathbf{?} \\ \mathit{c}\mathbf{)}\mathbf{=}\mathbf{5}\mathbf{?} \end{gathered}$$

Question: Find the probability of selecting exactly one of the correct six integers in a lottery, where the order in which these integers are selected does not matter, from the positive integers not exceeding

(a) 40.

(b) 48.

(c) 56.

(d) 64.

Question: What is the variance of the number of heads that come up when a fair coin is flipped 10 times?

Question: a) Solve this puzzle in two different ways. First, answer the problem by considering the probability of the gender of the second child. Then, determine the probability differently, by considering the four different possibilities for a family of two children.

b) Show that the answer to the puzzle becomes unambiguous if we also know that Mr. Smith chose his walking companion at random from his two children.

c) Another variation of this puzzle asks "When we meet Mr. Smith, he tells us that he has two children and at least one is a son. What is the probability that his other child is a son?' Solve this variation of the puzzle, explaining why it is unambiguous.

Question:In a super lottery, a player selects 7 numbers out of the first 80 positive integers. what is the probability that a person wins the grand prize by picking 7 numbers that are among the 11 numbers selected at random by a computer?

Question: What is the variance of the number of times a 6 appears when a fair die is rolled 10 times?

Question: Assume that the probability a child is a boy is 0.51 and that the sexes of the children born into a family are independent. What is the probability that a family of five children has

1. Exactly three boys?
2. At least one boy?
3. At least one girl?
4. All children of the same sex?

Question:In 2010, the puzzle designer Gary Foshee posed this problem: "Mr. Smith has two children, one of whom is a son born on a Tuesday. What is the probability that Mr. Smith has two sons?" Show that there are two different answers to this puzzle, depending on whether Mr. Smith specifically mentioned his son because has born on a Tuesday or whether he randomly chose a child and reported its gender and birth day of the week. (Hint: For the first possibility, enumerate all the equally likely possibilities for the gender and birth day of the week of the other child. To do, this consider first the cases where the older child is a boy born on a Tuesday and then the case where the older child is not a boy born on a Tuesday.).

Question: A group of six people play the game of “odd person out” to determine who will buy refreshments. Each person flips a fair coin. If there is a person whose outcome is not the same as that of any other member of the group, this person has to buy the refreshments. What is the Probability that there is an odd person out after the coins are flipped once?

Question:In a super lottery, a player wins a fortune if they choose the eight numbers selected by a computer from the positive integers not exceeding 100. What is the probability that a player wins this super lottery?