Problem 65 - Basic Statistics for Business and Economics with Student

Question

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 odds of winning a free soft drink are 1 in \(10,\) and the odds of winning a free large pizza are 1 in \(50 .\) 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?

Step-by-Step Solution

Answer

a. 12%; b. 88%; c. 68.2%; d. 31.8%.

Step 1: Understand the probabilities

The probability of winning a soft drink is \( \frac{1}{10} \) and the probability of winning a large pizza is \( \frac{1}{50} \). To find the probability of winning either prize, we add these probabilities because winning the prizes are mutually exclusive events.

Step 2: Calculate the probability of winning either prize

To find the probability of winning either a soft drink or a large pizza, we add the two probabilities: \[ P(\text{win a prize}) = \frac{1}{10} + \frac{1}{50} = \frac{5}{50} + \frac{1}{50} = \frac{6}{50} = \frac{3}{25} \approx 0.12 \] or 12%.

Step 3: Calculate the probability of not winning a prize on one visit

The probability of not winning any prize during a single visit is the complement of winning a prize. This is given by:\[ P(\text{not winning}) = 1 - P(\text{win a prize}) = 1 - \frac{3}{25} = \frac{22}{25} \approx 0.88 \] or 88%.

Step 4: Calculate the probability of not winning a prize on three consecutive visits

To find the probability of not winning a prize on three consecutive visits, multiply the probability of not winning on one visit raised to the power of 3:\[ P(\text{not winning on 3 visits}) = \left(\frac{22}{25}\right)^3 = \frac{10648}{15625} \approx 0.682 \] or 68.2%.

Step 5: Calculate the probability of winning at least one prize on the next three visits

The probability of winning at least one prize on one of your next three visits is the complement of not winning any prize at all across these visits:\[ P(\text{win at least one prize on 3 visits}) = 1 - \left(\frac{22}{25}\right)^3 = 1 - 0.682 = 0.318 \] or 31.8%.

Key Concepts

Complementary Probability

Complementary probability is all about understanding and calculating the likelihood of an event not happening. It's an important concept in probability as it allows us to look at the negative side of an event, which often complements our analysis of the positive side. This essentially means the probability of the event not occurring is equal to one minus the probability of the event occurring.
In the provided exercise, to find the probability of not winning any prize during a visit to Tony's Pizza, we first determined the probability of winning a prize. Since the probability of winning a prize is 12%, the complementary probability would be:

• Formula:
• \[ P( ext{not winning a prize}) = 1 - P( ext{win a prize}) \]

The result is an 88% chance of not winning. Complementary probability helps simplify complex scenarios by focusing on what is not expected to happen, which can often be easier to calculate and reason through.

Mutually Exclusive Events

Mutually exclusive events are situations where two or more events cannot occur at the same time. An easy way to remember this is by thinking of the term 'mutually exclusive' as never meeting: one event happening excludes the possibility of the other happening at the same time.
When calculating the probability of winning either the soft drink or the large pizza at Tony's Pizza, we rely on the concept of mutually exclusive events, since a customer can't win both prizes on a single purchase. Thus, to find the probability of winning one prize, we add the separate probabilities of each prize:

• Winning soft drink: \( \frac{1}{10} \)
• Winning large pizza: \( \frac{1}{50} \)

These added together derive the probability of winning either prize, ensuring a 12% (or \( \frac{3}{25} \)) chance.

Consecutive Probability Calculations

When we talk about consecutive probability calculations, we mean repeatedly calculating the probability of an event over several occasions. This often involves multiplying probabilities when dealing with repeated independent events.
In the exercise, we looked at the probability of not winning a prize over three consecutive visits to Tony's Pizza. To compute this, we used the probability that on each visit, no prize will be won. As each visit is an independent event, you'd multiply the probability of not winning on a single visit by itself for the number of visits desired:

• Non-winning probability for one visit: \( \frac{22}{25} \)
• Three consecutive visits: \( \left(\frac{22}{25}\right)^3 \)

This gives a 68.2% chance of not winning across three visits. Changing this to determine the chance of winning at least once is a matter of finding the complement of not winning across all visits. These calculations help us understand how likelihoods change over repeated events, emphasizing the power of consecutive measures in probability theory.