91Ó°ÊÓ

A random sample of 400 college students was asked if college athletes should be paid. The following table gives a two-way classification of the responses. $$\begin{array}{lcc} \hline & \text { Should Be Paid } & \text { Should Not Be Paid } \\ \hline \text { Student athlete } & 90 & 10 \\ \text { Student nonathlete } & 210 & 90 \end{array}$$ a. If one student is randomly selected from these 400 students, find the probability that this student i. is in favor of paying college athletes ii. favors paying college athletes given that the student selected is a nonathlete iii. is an athlete and favors paying student athletes iv. is a nonathlete or is against paying student athletes b. Are the events "student athlete" and "should be paid" independent? Are they mutually exclusive? Explain why or why not.

Step by step solution

01

Calculate the probability of a student favoring pay for athletes.

For part i, to find the probability that a randomly selected student supports the payment of college athletes, add the number of student athletes and non-athletes who support it. This will be divided by the total number of students. The calculation is therefore \((90 + 210) / 400 = 0.75\).

02

Determine the probability of a student supporting payment of athletes, given student is a non-athlete.

For part ii, to calculate the probability that a student supports paying athletes given that the student is a non-athlete, divide the count of non-athletes who support the payment by the total number of non-athletes. The calculation is \((210) / (210+90) = 0.7\).

03

Probability Calculation for Student Athlete Who Supports Payment.

For part iii, to find the probability that a student is an athlete and supports the payment of athletes, divide the number of student athletes who support payment by the total number of students. The calculation is \((90) / (400) = 0.225\).

04

Calculate the probability that a student is a non-athlete or against paying athletes.

For part iv, to find the probability that a student is a non-athlete or against the payment of athletes, add the number of non-athletes and those who are against payment, then divide by the total number of students. The calculation is therefore \((210+90 + 10) / (400) = 0.775\).

05

Determine Independence and Mutual Exclusivity of Events.

For part b, two events A and B are said to be independent if the occurrence of one does not affect the occurrence of the other. In this case, being a student athlete (event A) seems not to affect the likelihood of supporting payment for athletes (event B), given the probabilities calculated in parts i and ii. \n However, to be sure, we can test it. If A and B are independent, then \(P(A and B)\) should equal \(P(A)P(B)\). We calculated earlier that \(P(A and B)\) as 0.225, while \(P(A)\) and \(P(B)\) can be found as \(100/400 = 0.25\) and \(300/400 = 0.75\) respectively. Since \(0.225 \neq 0.25 \cdot 0.75\), the events are not independent. Also, since it's possible to be a student athlete and support the payment of students, the two events are not mutually exclusive.

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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 the probability of one event occurring given that another event has already happened. It's like knowing more specific info to better predict outcomes.
In our exercise, a conditional probability example was calculating the chance a student supports athlete payment, knowing the student is a non-athlete.
We defined the conditional probability of an event A given B as: \[P(A \mid B) = \frac{P(A \cap B)}{P(B)}\]where:

• \(P(A \cap B)\) is the probability of both events occurring.
• \(P(B)\) is the probability that event B occurs.

This way, we can find that 70% of non-athletes support payment, giving us insights based on specific criteria.

Independence of Events

Independence of events describes situations where the occurrence of one event doesn't affect the probability of another. Think of two unconnected situations where the outcome of one does not impact the other.
To figure out if events A and B are independent, you check if \[P(A \cap B) = P(A) \cdot P(B)\]If this holds true, the events don't affect each other. In our exercise, we calculated:

• Probability of 'student athlete' and 'should be paid' = 0.225
• Probability of 'student athlete" = 0.25
• Probability of"should be paid" = 0.75

Since 0.225 isn’t equal to 0.25 \(\cdot\) 0.75, they're not independent. Understanding such relationships is key in probability.

Mutually Exclusive Events

Mutually exclusive events refer to situations where two events can't occur at the same time. If one happens, the other simply can't.
For instance, flipping a coin to land heads or tails. If it’s heads, it can't be tails simultaneously.
In our given exercise, we found that being a student athlete and supporting payment aren’t mutually exclusive. Both can happen since a student athlete can also support paying athletes.
In probability terms, mutually exclusive events A and B mean\[P(A \cap B) = 0\]If there’s any scenario where both occur together, they’re not mutually exclusive, just like our example.

Two-Way Tables

Two-way tables, or contingency tables, are fundamental tools in organizing statistics data. They help categorize data by groups, enabling better analysis and probability calculations.
In our exercise with a two-way table, it categorizes participants by athletic status and payment support.
This format helps visualize relationships between variables and perform key calculations. Two-way tables allow for easier breakdown of conditional probabilities, total probabilities, and exploring event independence.

• Find totals in rows or columns to get probabilities quickly.
• Compare groups efficiently, thanks to clear data organization.

Using two-way tables simplifies seeing how different groups relate, offering insights that can inform decisions or further analysis. They’re an effective method for organizing categorical data and conducting probability evaluations.