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A company that plans to hire one new employee has prepared a final list of six candidates, all of whom are equally qualified. Four of these six candidates are women. If the company decides to select at random one person out of these six candidates, what is the probability that this person will be a woman? What is the probability that this person will be a man? Do these two probabilities add up to \(1.0 ?\) If yes, why?

Briefly explain the difference between the marginal and conditional probabilities of events. Give one example of each.

What is meant by two mutually exclusive events? Give one example of two mutually exclusive events and another example of two mutually nonexclusive events.

A statistical experiment has eight equally likely outcomes that are denoted by \(1,2,3,4,5,6,7\), and 8\. Let event \(A=\\{2,5,7\\}\) and event \(B=\\{2,4,8\\}\). a. Are events \(A\) and \(B\) mutually exclusive events? b. Are events \(A\) and \(B\) independent events? c. What are the complements of events \(A\) and \(B\), respectively, and their probabilities?

A small ice cream shop has 10 flavors of ice cream and 5 kinds of toppings for its sundaes. How many different selections of one flavor of ice cream and one kind of topping are possible?

A man just bought 4 suits, 8 shirts, and 12 ties. All of these suits, shirts, and ties coordinate with each other. If he is to randomly select one suit, one shirt, and one tie to wear on a certain day, how many different outcomes (selections) are possible?

Two thousand randomly selected adults were asked whether or not they have ever shopped on the Internet. The following table gives a two-way classification of the responses. $$\begin{array}{lcc} \hline & \text { Have Shopped } & \text { Have Never Shopped } \\ \hline \text { Male } & 500 & 700 \\ \text { Female } & 300 & 500 \end{array}4$ a. If one adult is selected at random from these 2000 adults, find the probability that this adult i. has never shopped on the Internet ii. is a male iii. has shopped on the Internet given that this adult is a female iv. is a male given that this adult has never shopped on the Internet b. Are the events "male" and "female" mutually exclusive? What about the events "have shopped" and "male?" Why or why not? c. Are the events "female" and "have shopped" independent? Why or why not?

A consumer agency randomly selected 1700 flights for two major airlines, \(\mathrm{A}\) and \(\mathrm{B}\). The following table gives the two-way classification of these flights based on airline and arrival time. Note that "less than 30 minutes late" includes flights that arrived early or on time. $$\begin{array}{lccc} \hline & \begin{array}{c} \text { Less Than 30 } \\ \text { Minutes Late } \end{array} & \begin{array}{c} \text { 30 Minutes to } \\ \text { 1 Hour Late } \end{array} & \begin{array}{c} \text { More Than } \\ \text { 1 Hour Late } \end{array} \\ \hline \text { Airline A } & 429 & 390 & 92 \\ \text { Airline B } & 393 & 316 & 80 \\ \hline \end{array}$$ a. If one flight is selected at random from these 1700 flights, find the probability that this flight is \(\mathrm{i}\), more than 1 hour late ii. less than 30 minutes late iii. a flight on airline A given that it is 30 minutes to 1 hour late iv. more than 1 hour late given that it is a flight on airline \(\mathrm{B}\) b. Are the events "airline A" and "more than 1 hour late" mutually exclusive? What about the events "less than 30 minutes late" and "more than 1 hour late?" Why or why not? c. Are the events "airline \(\mathrm{B}\) " and " 30 minutes to 1 hour late" independent? Why or why not?

Define the following two events for two tosses of a coin: \(A=\) at least one head is obtained \(B=\) both tails are obtained a. Are \(A\) and \(B\) mutually exclusive events? Are they independent? Explain why or why not. b. Are \(A\) and \(B\) complementary events? If yes, first calculate the probability of \(B\) and then calculate the probability of \(A\) using the complementary event rule.

Let \(A\) be the event that a number less than 3 is obtained if we roll a die once. What is the probability of \(A ?\) What is the complementary event of \(A\), and what is its probability?