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Chapter 4: Problem 27

A 2001 preseason NCAA football poll asked respondents to answer the question, "Will the Big Ten or the Pac-10 have a team in this year's national championship game, the Rose Bowl?" Of the 13,429 respondents, 2961 said the Big Ten would, 4494 said the Pac- 10 would, and 6823 said neither the Big Ten nor the Pac- 10 would have a team in the Rose Bowl (http://www.yahoo.com, August 30, 2001). a. What is the probability that a respondent said neither the Big Ten nor the Pac-10 would have a team in the Rose Bowl? b. What is the probability that a respondent said either the Big Ten or the Pac-10 would have a team in the Rose Bowl? c. Find the probability that a respondent said both the Big Ten and the Pac-10 would have a team in the Rose Bowl.

Short Answer

Expert verified
a. \( P(\text{Neither}) = \frac{6823}{13429} \) b. \( P(\text{Either}) = 1 - \frac{6823}{13429} \) c. \( P(\text{Both}) = \frac{849}{13429} \)

Step by step solution

01

Calculate Total Respondents

The total number of respondents in the survey is given as 13,429. This number will be used as the denominator in all probability calculations as it represents the total sample space.
02

Calculate Probability of 'Neither'

To find the probability that a respondent said neither the Big Ten nor the Pac-10 would have a team in the Rose Bowl, divide the number of respondents who said neither (6,823) by the total number of respondents (13,429). The probability is given by: \[ P(\text{Neither}) = \frac{6823}{13429} \]
03

Calculate Probability of 'Either Big Ten or Pac-10'

To calculate the probability that a respondent said either the Big Ten or the Pac-10 (or both) would have a team, use the complement rule. Add the individual probabilities of Big Ten and Pac-10 each having a team, and subtract the probability that both have a team: \[ P(\text{Either}) = \left(\frac{2961}{13429} + \frac{4494}{13429} \right) - P(\text{Both}) \] Where \( P(\text{Both}) \) can be found in Step 4.
04

Use Complement to Find 'Both Big Ten and Pac-10'

To find the probability that both the Big Ten and Pac-10 have a team in the Rose Bowl, use the given information: the complement of those who said neither is those who said either or both. Let total saying either or both be \( T_{\text{Either}} = 13,429 - 6,823 = 6,606 \). Knowing that this includes either of them and both, the intersection (both) would be: \[ P(\text{Both}) = \frac{2961 + 4494 - 6606}{13429} \] Simplified, \( 2961 + 4494 - 6606 = 849 \), thus \[ P(\text{Both}) = \frac{849}{13429} \].

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Key Concepts

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

Statistical Probability
Probability is a crucial concept in statistics, helping us understand and quantify uncertainty. When we talk about statistical probability, we're talking about the likelihood of an event happening based on collected data. In the context of a survey, like the one from the NCAA preseason poll, this involves looking at how many respondents selected a particular option out of the total surveyed.

Probability is calculated as a fraction or percentage using the formula:
- Probability = (Number of favorable outcomes) / (Total number of outcomes)

For example, to find the probability that respondents said neither the Big Ten nor the Pac-10 would play in the Rose Bowl, we used:
- Probability of `Neither` =\( \frac{6823}{13429} \)
This equates to roughly 0.508, meaning about 50.8% of the respondents believed neither team would make it to the Rose Bowl.

Understanding statistical probability helps in interpreting survey results and making informed predictions.
Survey Analysis
Survey analysis involves examining the outcomes of collecting data through surveys. It allows us to draw conclusions from responses, predict trends, and inform decisions. Each response adds to the overall picture, showing popular opinions or common beliefs.

In the given survey about NCAA football, we collected responses regarding two key possibilities: whether the Big Ten, the Pac-10, or neither would feature in the Rose Bowl. By breaking down the responses:
  • 2,961 believed the Big Ten would be in the game.
  • 4,494 thought the Pac-10 would make it.
  • 6,823 believed neither would have a team in the final.
This data can then be used to calculate probabilities, which aids in drawing meaningful insights.

Effective survey analysis requires clear questions, a representative sample size, and logical interpretation鈥攕kills necessary to ensure reliable conclusions.
Complement Rule
The complement rule is a staple in probability theory. It helps find the probability of an event by calculating the likelihood of the opposite event. In simple terms, if you know the probability of something not happening (the complement), you can easily deduce the probability of it happening.

Mathematically, if the probability of event A is 0.7, the probability of the complement (not A) is 1 - 0.7, or 0.3.
  • This rule simplifies calculations by providing an alternative route to probability determination.

In our football poll survey, we used the complement rule to determine the probability of either the Big Ten or Pac-10 team being in the Rose Bowl. While we had direct numbers for each conference, finding the 'both' probability required recognizing that those who said neither would be the complement of those selecting either or both. Hence:
- Probability of `Either` = 1 - Probability of `Neither`
This method efficiently turns our raw data into actionable insight, all thanks to clever use of mathematical principles.

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Most popular questions from this chapter

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?

The prior probabilities for events \(A_{1}, A_{2},\) and \(A_{3}\) are \(P\left(A_{1}\right)=.20, P\left(A_{2}\right)=.50,\) and \(P\left(A_{3}\right)=\) \(.30 .\) The conditional probabilities of event \(B\) given \(A_{1}, A_{2},\) and \(A_{3}\) are \(P\left(B | A_{1}\right)=.50\) \(P\left(B | A_{2}\right)=.40,\) and \(P\left(B | A_{3}\right)=.30\) a. Compute \(P\left(B \cap A_{1}\right), P\left(B \cap A_{2}\right),\) and \(P\left(B \cap A_{3}\right)\) b. Apply Bayes' theorem, equation (4.19), to compute the posterior probability \(P\left(A_{2} | B\right)\). c. Use the tabular approach to applying Bayes' theorem to compute \(P\left(A_{1} | B\right), P\left(A_{2} | B\right)\) and \(P\left(A_{3} | B\right)\)

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.

How many ways can three items be selected from a group of six items? Use the letters \(\mathrm{A}, \mathrm{B}\) \(\mathrm{C}, \mathrm{D}, \mathrm{E},\) and \(\mathrm{F}\) to identify the items, and list each of the different combinations of three items.

An experiment has three steps with three outcomes possible for the first step, two outcomes possible for the second step, and four outcomes possible for the third step. How many experimental outcomes exist for the entire experiment?
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