91Ó°ÊÓ

There is a \(15 \%\) chance that a shopper entering a computer store will purchase a computer, a \(25 \%\) chance they will purchase a game/software, and there is a \(10 \%\) chance they will purchase both a computer and a game/software. a. Create a contingency table for the information. b. What is the probability that a shopper will not purchase a computer and will not purchase a game/software? c. What is the probability that a shopper will purchase a computer or purchase a game/software? d. What is the probability that a shopper will purchase a game/software given they have purchased a computer? e. What is the probability that a shopper will purchase a game/software given they did not purchase a computer? f. Does it appear that purchasing a game/software depends on whether the shopper purchased a computer? Or are they independent? Use probability to support your claim.

Step by step solution

01

Define Probabilities

Let event \( C \) be that a shopper purchases a computer, and \( G \) be that a shopper purchases a game/software. We know: \( P(C) = 0.15 \), \( P(G) = 0.25 \), \( P(C \cap G) = 0.10 \).

02

Create the Contingency Table

A contingency table shows the probabilities for all combinations of events. Fill out the table:- Purchases a Computer and Game/Software: \( P(C \cap G) = 0.10 \)- Purchases only a Computer: \( P(C) - P(C \cap G) = 0.15 - 0.10 = 0.05 \)- Purchases only Game/Software: \( P(G) - P(C \cap G) = 0.25 - 0.10 = 0.15 \)- Purchases neither: \( 1 - (0.05 + 0.10 + 0.15) = 0.70 \) Contingency Table:|| Purchase Game/Software | No Purchase ||---|---|---|| Purchase Computer | 0.10 | 0.05 || No Purchase | 0.15 | 0.70 |

03

Probability of No Purchase

To find the probability that a shopper will not purchase a computer and will not purchase a game/software, use the table calculation: \( P(\text{No Computer and No Game/Software}) = 0.70 \)

04

Probability of Purchasing a Computer or Game/Software

Use the formula for the union of two events: \[ P(C \cup G) = P(C) + P(G) - P(C \cap G) \]Substitute the values:\[ P(C \cup G) = 0.15 + 0.25 - 0.10 = 0.30 \]

05

Probability of Game/Software Given Computer

Use the conditional probability formula:\[ P(G|C) = \frac{P(C \cap G)}{P(C)} \]Substitute the values:\[ P(G|C) = \frac{0.10}{0.15} = \frac{2}{3} \]

06

Probability of Game/Software Given No Computer

Use the conditional probability formula:\[ P(G|\text{No } C) = \frac{P(G) - P(C \cap G)}{1 - P(C)} \]Substitute the values:\[ P(G|\text{No } C) = \frac{0.15}{0.85} \approx 0.176 \]

07

Check for Independence

Two events are independent if \( P(G|C) = P(G) \). Since:\[ P(G|C) = \frac{2}{3} \quad\text{and}\quad P(G) = 0.25 \]These probabilities are not equal, indicating that purchasing a game/software depends on whether a computer is purchased.

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Key Concepts

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

Probability Theory

Probability theory is the mathematical framework that helps us understand random events and their outcomes. It enables us to calculate the likelihood of different events occurring, which we denote with probabilities. In this context, probabilities fall between 0 and 1, where 0 means an impossible event and 1 means a certain event. For example, the probability of a shopper purchasing a computer in our exercise is given as 0.15, indicating a 15% chance.

You can often express the probability of events in simpler terms using the concept of sets:

• If you have an event happening, like purchasing a game/software, the probability is denoted by \( P(G) \). In this case, it's 0.25, or 25%.
• Probabilities can be combined to evaluate the likelihood of multiple events occurring, such as both purchasing a computer and game/software, expressed as \( P(C \cap G) \). Here, that probability is 0.10, or 10%.
• The probability of not occurring is denoted as \( 1 - \text{Probability of occurring} \). This helps to find the chance that neither event happens, which in the contingency table equals 0.70, or 70%.

Conditional Probability

Conditional probability considers the probability of an event occurring, given that another event has already occurred. It's useful for understanding the relationships between two dependent events.

One way to calculate conditional probability is to use the formula: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]

In our scenario, if we want to know the probability of a shopper buying game/software given they have already purchased a computer, it is calculated as:

• The formula becomes: \( P(G|C) = \frac{P(C \cap G)}{P(C)} \)
• Inputting our values: \( \frac{0.10}{0.15} = \frac{2}{3} \). So, there is approximately a 66.67% chance of purchasing game/software if a computer was bought.

Additionally, the conditional probability when no computer is purchased is calculated slightly differently:

• It's given by: \( P(G|\text{No } C) = \frac{P(G) - P(C \cap G)}{1 - P(C)} \)
• Thus: \( \frac{0.15}{0.85} \approx 0.176 \). This means there's approximately a 17.6% chance of buying game/software even without purchasing a computer.

Independence in Probability

In probability, two events are considered independent if the outcome of one does not affect the outcome of another. To check for independence, you can use the conditional probability:

• Events are independent if: \( P(A|B) = P(A) \).

In our problem, to determine if purchasing game/software is independent of purchasing a computer, we compare:

• Calculated \( P(G|C) = \frac{2}{3} \) versus original \( P(G) = 0.25 \).

These probabilities differ; hence, in our example, purchasing game/software depends on whether a shopper also purchases a computer. This demonstrates dependency, illustrating that knowing one event occurs gives additional information about the likelihood of the other event occurring.