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Chapter 4: Problem 38

A Morgan Stanley Consumer Research Survey sampled men and women and asked each whether they preferred to drink plain bottled water or a sports drink such as Gatorade or Propel Fitness water (The Atlanta Journal-Constitution, December 28,2005 ). Suppose 200 men and 200 women participated in the study, and 280 reported they preferred plain bottled water. Of the group preferring a sports drink, 80 were men and 40 were women. Let \(M=\) the event the consumer is a man \(W=\) the event the consumer is a woman \(B=\) the event the consumer preferred plain bottled water \(S=\) the event the consumer preferred sports drink a. What is the probability a person in the study preferred plain bottled water? b. What is the probability a person in the study preferred a sports drink? c. What are the conditional probabilities \(P(M | S)\) and \(P(W | S) ?\) d. What are the joint probabilities \(P(M \cap S)\) and \(P(W \cap S) ?\) e. Given a consumer is a man, what is the probability he will prefer a sports drink? f. Given a consumer is a woman, what is the probability she will prefer a sports drink? g. Is preference for a sports drink independent of whether the consumer is a man or a woman? Explain using probability information.

Short Answer

Expert verified
a) 0.7, b) 0.3, c) 2/3, 1/3, d) 0.2, 0.1, e) 0.4, f) 0.2, g) Preferences are not independent.

Step by step solution

01

Calculate Total Number of People

The study includes both men and women, each amounting to 200. Therefore, the total number of participants is:\[200 \text{ men} + 200 \text{ women} = 400 \text{ people}.\]
02

Probability of Preferring Bottled Water

To find the probability of a person preferring plain bottled water, use the formula for probability:\[P(B) = \frac{\text{Number of people preferring bottled water}}{\text{Total number of people}} = \frac{280}{400} = 0.7.\]
03

Probability of Preferring a Sports Drink

Use the number of people that prefer sports drinks (80 men + 40 women = 120 people) to calculate the probability:\[P(S) = \frac{120}{400} = 0.3.\]
04

Conditional Probability for Men Given Sports Drink

To calculate the probability that a person is a man, given that they prefer sports drinks:\[P(M | S) = \frac{\text{Number of men preferring sports drinks}}{\text{Total number preferring sports drinks}} = \frac{80}{120} = \frac{2}{3}.\]
05

Conditional Probability for Women Given Sports Drink

To calculate the probability that a person is a woman, given that they prefer sports drinks:\[P(W | S) = \frac{\text{Number of women preferring sports drinks}}{\text{Total number preferring sports drinks}} = \frac{40}{120} = \frac{1}{3}.\]
06

Joint Probability for Men and Sports Drink

Calculate the joint probability that a person is both a man and prefers a sports drink:\[P(M \cap S) = \frac{\text{Number of men preferring sports drinks}}{\text{Total number of people}} = \frac{80}{400} = 0.2.\]
07

Joint Probability for Women and Sports Drink

Calculate the joint probability that a person is both a woman and prefers a sports drink:\[P(W \cap S) = \frac{\text{Number of women preferring sports drinks}}{\text{Total number of people}} = \frac{40}{400} = 0.1.\]
08

Conditional Probability for Sports Drink Given Man

Given a person is a man, the probability he prefers sports drinks is:\[P(S | M) = \frac{\text{Number of men preferring sports drinks}}{\text{Total number of men}} = \frac{80}{200} = 0.4.\]
09

Conditional Probability for Sports Drink Given Woman

Given a person is a woman, the probability she prefers sports drinks is:\[P(S | W) = \frac{\text{Number of women preferring sports drinks}}{\text{Total number of women}} = \frac{40}{200} = 0.2.\]
10

Check for Independence Using Probabilities

To determine if preference for a sports drink is independent of being a man or a woman, check if:\[P(S | M) = P(S) \quad \text{and} \quad P(S | W) = P(S).\]Since \(P(S | M) = 0.4\) and \(P(S) = 0.3\) are not equal, and \(P(S | W) = 0.2\) and \(P(S) = 0.3\) are not equal, preference for a sports drink is not independent of being a man or a woman.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conditional Probability
Conditional probability helps us understand how to calculate the likelihood of an event occurring, given that another event has already occurred. In other words, it tells us the probability of a particular outcome, provided some information about another related event. For example, if you want to know the probability that a person is a woman, given that they prefer a sports drink, you would use conditional probability.
The formula for calculating conditional probability is: \[P(A | B) = \frac{P(A \cap B)}{P(B)}\] Where,
  • \(P(A | B)\): Probability of event A happening given that B has happened.
  • \(P(A \cap B)\): Joint probability of both events A and B occurring.
  • \(P(B)\): Probability that event B occurs.
In our exercise, the probability of a person being a woman given they prefer sports drinks, \(P(W | S)\), is calculated by observing how many of those who prefer sports drinks are women and dividing it by the total number who prefer sports drinks.
This approach provides a fine-tuned understanding of probabilities in contexts where related conditions are known.
Joint Probability
Joint probability refers to the likelihood of two events happening at the same time. It gives us insight into the overlap or intersection of two events in a probability space. If you have two events, such as preferring a sports drink and being a man, joint probability helps determine the chance that both are true simultaneously.
To find joint probability, you use the formula: \[P(A \cap B) = \frac{\text{Number of favorable outcomes for both A and B}}{\text{Total number of possible outcomes}}\] In our scenario, this would be figuring out how many men prefer sports drinks compared to the total number of participants.
  • Practical Use: Understanding joint probabilities is essential when dealing with multiple events and helps in statistics and predictions for longer-term trends.
  • Calculation: It requires counting the intersection of the two events over the total number of cases considered. This is quite straightforward with clear data.
Thus, joint probability is a fundamental concept for assessing the probability of related events occurring together within a data set.
Independence in Probability
Independence in probability occurs when two events do not affect each other's likelihood of occurring. This implies that gaining information about one event doesn't shed any light on the occurrence of the other.
Two events A and B are independent if: \[P(A | B) = P(A)\] \[\text{and}\] \[P(B | A) = P(B)\] This means that knowing B occurs does not change the probability of A occurring, and vice versa. In practical terms, this is like saying whether someone prefers sports drinks doesn鈥檛 change whether they are a man or a woman, and vice versa.
  • Assessment of Independence: To determine if preference for a sports drink is independent of gender in our study, compare \(P(S | M)\) and \(P(S | W)\) with \(P(S)\).
  • Example: In the exercise, since \(P(S | M)\) and \(P(S | W)\) do not match \(P(S)\), we conclude that the preference for sports drinks is indeed dependent on gender.
Recognizing independence helps simplify complex probability models by identifying unrelated elements.

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Most popular questions from this chapter

Clarkson University surveyed alumni to learn more about what they think of Clarkson. One part of the survey asked respondents to indicate whether their overall experience at Clarkson fell short of expectations, met expectations, or surpassed expectations. The results showed that \(4 \%\) of the respondents did not provide a response, \(26 \%\) said that their experience fell short of expectations, and \(65 \%\) of the respondents said that their experience met expectations (Clarkson Magazine, Summer 2001). a. If we chose an alumnus at random, what is the probability that the alumnus would say their experience surpassed expectations? b. If we chose an alumnus at random, what is the probability that the alumnus would say their experience met or surpassed expectations?

An oil company purchased an option on land in Alaska. Preliminary geologic studies assigned the following prior probabilities. \\[ \begin{aligned} P(\text { high-quality oil }) &=.50 \\ P(\text { medium-quality oil }) &=.20 \\ P(\text { no oil }) &=.30 \end{aligned} \\] a. What is the probability of finding oil? b. After 200 feet of drilling on the first well, a soil test is taken. The probabilities of finding the particular type of soil identified by the test follow. \\[ \begin{aligned} P(\text { soil } | \text { high-quality oil }) &=.20 \\ P(\text { soil } | \text { medium-quality oil }) &=.80 \\ P(\text { soil } | \text { no oil }) &=.20 \end{aligned} \\] How should the firm interpret the soil test? What are the revised probabilities, and what is the new probability of finding oil?

To investigate how often we eat at home as a family during the week, Harris Interactive surveyed 496 adults living with children under the age of \(18(U S A \text { Today, January } 3,2007\) ). The survey results are shown in the following table. $$\begin{array}{cc}\text { Number of } & \text { Number of Times } \\ \text { Family Meals } & \text { Outcome Occurred } \\ 0 & 11 \\ 1 & 11 \\ 2 & 30 \\\ 3 & 36 \\ 4 & 36 \\ 5 & 119 \\ 6 & 114 \\ 7 \text { or more } & 139\end{array}$$ For a randomly selected family with children under the age of \(18,\) compute the following: a. The probability the family eats no meals at home during the week. b. The probability the family eats at least four meals at home during the week. c. The probability the family eats two or fewer meals at home during the week.

The National Sporting Goods Association conducted a survey of persons 7 years of age or older about participation in sports activities (Statistical Abstract of the United States: 2002 ). The total population in this age group was reported at 248.5 million, with 120.9 million male and 127.6 million female. The number of participants for the top five sports activities appears here $$\begin{array}{l|cc} & \multicolumn{2}{c} {\text { Participants (millions) }} \\ \text { Activity } & \text { Male } & \text { Female } \\ \hline \text { Bicycle riding } & 22.2 & 21.0 \\ \text { Camping } & 25.6 & 24.3 \\ \text { Exercise walking } & 28.7 & 57.7 \\ \text { Exercising with equipment } & 20.4 & 24.4 \\ \text { Swimming } & 26.4 & 34.4 \end{array}$$ a. For a randomly selected female, estimate the probability of participation in each of the sports activities. b. For a randomly selected male, estimate the probability of participation in each of the sports activities. c. For a randomly selected person, what is the probability the person participates in exercise walking? d. Suppose you just happen to see an exercise walker going by. What is the probability the walker is a woman? What is the probability the walker is a man?

A large consumer goods company ran a television advertisement for one of its soap products. On the basis of a survey that was conducted, probabilities were assigned to the following events. \(B=\) individual purchased the product \(S=\) individual recalls seeing the advertisement \(B \cap S=\) individual purchased the product and recalls seeing the advertisement The probabilities assigned were \(P(B)=.20, P(S)=.40,\) and \(P(B \cap S)=.12\) a. What is the probability of an individual's purchasing the product given that the individual recalls seeing the advertisement? Does seeing the advertisement increase the probability that the individual will purchase the product? As a decision maker, would you recommend continuing the advertisement (assuming that the cost is reasonable)? b. Assume that individuals who do not purchase the company's soap product buy from its competitors. What would be your estimate of the company's market share? Would you expect that continuing the advertisement will increase the company's market share? Why or why not? c. The company also tested another advertisement and assigned it values of \(P(S)=.30\) and \(P(B \cap S)=.10 .\) What is \(P(B | S)\) for this other advertisement? Which advertisement seems to have had the bigger effect on customer purchases?
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