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Chapter 5: Problem 65

With each purchase of a large pizza at Tony's Pizza, the customer receives a coupon that can be scratched to see if a prize will be awarded. The probability of winning a free soft drink is \(0.10,\) and the probability of winning a free large pizza is \(0.02 .\) You plan to eat lunch tomorrow at Tony's. What is the probability: a. That you will win either a large pizza or a soft drink? b. That you will not win a prize? c. That you will not win a prize on three consecutive visits to Tony's? d. That you will win at least one prize on one of your next three visits to Tony's?

Short Answer

Expert verified
a. 0.12, b. 0.88, c. 0.6815, d. 0.3185

Step by step solution

01

Define Probability Terms

Let's first identify the given probabilities: - The probability of winning a free soft drink when you purchase a pizza, denoted as \( P(S) \), is \( 0.10 \). - The probability of winning a free large pizza, denoted as \( P(L) \), is \( 0.02 \). Assuming that winning a soft drink and a pizza are mutually exclusive events, the probability of winning either a soft drink or a pizza is the sum of their individual probabilities.
02

Calculate Probability of Winning a Prize

To find the probability of winning either a large pizza or a soft drink, we add the two probabilities: \[P(S \text{ or } L) = P(S) + P(L) = 0.10 + 0.02 = 0.12\]
03

Calculate Probability of Not Winning a Prize

The probability of not winning a prize is the complement of winning a prize. Given that \( P(S \text{ or } L) = 0.12 \), the probability of not winning any prize is:\[P(\text{no prize}) = 1 - P(S \text{ or } L) = 1 - 0.12 = 0.88\]
04

Probability of Not Winning a Prize on Three Consecutive Visits

If the probability of not winning a prize on one visit is \( 0.88 \), then the probability of not winning any prize on three consecutive visits is:\[P(\text{no prize in 3 visits}) = (0.88)^3 \] Calculate this as:\[P(\text{no prize in 3 visits}) \approx 0.681472\]
05

Probability of Winning at Least One Prize in Three Visits

The probability of winning at least one prize in three visits is the complement of not winning any prize in all three visits:\[P(\text{at least one prize in 3 visits}) = 1 - P(\text{no prize in 3 visits})\] Substitute the value from Step 4:\[P(\text{at least one prize in 3 visits}) = 1 - 0.681472 \approx 0.318528\]

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Key Concepts

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

Complement Rule
In probability theory, the Complement Rule is an essential tool. It is used to find the probability that an event does not occur. Simply put, if the probability of an event happening is known, the probability of it not happening is its complement. This is calculated using the formula:\[P( ext{not } A) = 1 - P(A)\]For example, if the probability of winning a prize is 0.12, the probability of not winning a prize is:
  • Calculate the complement: \(P( ext{no prize}) = 1 - 0.12 = 0.88\)
The rule efficiently helps us find the missing piece of the probability puzzle.
Mutually Exclusive Events
Understanding mutually exclusive events is crucial when analyzing probabilities. Mutually exclusive events cannot happen at the same time. This means if one event occurs, the other cannot. For example, winning a free drink and winning a large pizza are mutually exclusive at Tony's Pizza.To find the probability that one of two mutually exclusive events will occur, simply add their probabilities:
  • Formula: \(P(A \text{ or } B) = P(A) + P(B)\)
  • Example: \(P(S \text{ or } L) = 0.10 + 0.02 = 0.12\)
This straightforward addition works because the two events cannot overlap or occur simultaneously.
Probability Calculation
Calculating probability involves understanding the ratio of successful outcomes to the total number of possible outcomes. In scenarios with clear events, such as at Tony鈥檚 Pizza, this calculation becomes direct.
  • Winning a free soft drink: Probability of 0.10.
  • Winning a large pizza: Probability of 0.02.
  • Using probabilities for combined events, like winning any prize, you can use the sum for mutually exclusive events: \(0.10 + 0.02 = 0.12\).
These probability calculations empower you to predict outcomes effectively, enabling decisions based on likelihoods rather than guesses.
Consecutive Events
Consecutive events involve finding the probability of an event occurring over multiple successive trials. For this, multiplication is key, assuming each event is independent. The chance of not winning a prize on consecutive visits to Tony's Pizza is found as follows:If the probability of not winning on one visit is 0.88, then for three visits:
  • \(P( ext{no prize in 3 visits}) = 0.88^3\)
  • Calculation: \(0.88 \times 0.88 \times 0.88 \approx 0.681472\)
For at least one prize in these three visits, subtract from 1:
  • \(P( ext{at least one prize}) = 1 - 0.681472 \approx 0.318528\)
This approach allows you to assess probabilities over time, providing a clear understanding of outcomes in successive trials.

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Most popular questions from this chapter

A local bank reports that \(80 \%\) of its customers maintain a checking account, \(60 \%\) have a savings account, and \(50 \%\) have both. If a customer is chosen at random, what is the probability the customer has either a checking or a savings account? What is the probability the customer does not have either a checking or a savings account?

Several years ago, Wendy's advertised that there are 256 different ways to order your hamburger. You may choose to have, or omit, any combination of the following on your hamburger: mustard, ketchup, onion, pickle, tomato, relish, mayonnaise, and lettuce. Is the advertisement correct? Show how you arrive at your answer.

The board of directors of Saner Automatic Door Company consists of 12 members, 3 of whom are women. A new policy and procedures manual is to be written for the company. A committee of three is randomly selected from the board to do the writing. a. What is the probability that all members of the committee are men? b. What is the probability that at least one member of the committee is a woman?

There are four people being considered for the position of chief executive officer of Dalton Enterprises. Three of the applicants are over 60 years of age. Two are female, of which only one is over \(60 .\) a. What is the probability that a candidate is over 60 and female? b. Given that the candidate is male, what is the probability he is younger than \(60 ?\) c. Given that the person is over \(60,\) what is the probability the person is female?

Suppose the probability you will get an \(\mathrm{A}\) in this class is .25 and the probability you will get a \(\mathrm{B}\) is . \(50 .\) What is the probability your grade will be above a C?
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