91Ó°ÊÓ

A random sample of 250 adults was taken, and they were asked whether they prefer watching sports or opera on television. The following table gives the two-way classification of these adults $$\begin{array}{lcc} \hline & \begin{array}{c} \text { Prefer Watching } \\ \text { Sports } \end{array} & \begin{array}{c} \text { Prefer Watching } \\ \text { Opera } \end{array} \\ \hline \text { Male } & 96 & 24 \\ \text { Female } & 45 & 85 \end{array}$$ a. If one adult is selected at random from this group, find the probability that this adult i. prefers watching opera ii. is a male iii. prefers watching sports given that the adult is a female iv. is a male given that he prefers watching sports \(\mathbf{v}\). is a female and prefers watching opera vi. prefers watching sports or is a male b. Are the events "female" and "prefers watching sports" independent? Are they mutually exclusive? Explain why or why not.

Step by step solution

01

Define Total and Individual Numbers

The total number of adults surveyed is 250. Males who prefer sports is 96, males who prefer opera is 24, females who prefer sports is 45 and females who prefer opera is 85. Total number of individuals who prefer sports = 96(males) + 45(females) = 141. Total number of individuals who prefer opera= 24(males) + 85(females) = 109. Total males = 96(sports) + 24(opera) =120 . Total females = 45(Sports) + 85(opera) = 130 .

02

Calculate Probabilities (a i-vi)

i. Probability of randomly selecting someone who prefers watching opera\( P(O)\) = total opera watchers / total surveyed = 109/250 = 0.436. ii. Probability of randomly selecting a male \( P(M) \) = total males / total surveyed = 120/250 = 0.48. iii. Probability of a female preferring sports \( P(S|F) \) = females who watch sports / total females = 45/130 = 0.346. iv. Probability of a male given he watches sports \( P(M|S) \) = males who watch sports / total who watch sports = 96/141 = 0.681. v. Probability of a female preferring opera \(P(FO)\) = females who watch opera / total surveyed = 85/250 = 0.34. vi. Probability of a male or someone who watches sports \( P(M ∪ S) \) = P(M) + P(S) - P(M ∩ S) = 0.48 + 0.564 - 0.384 = 0.66.

03

Check for Independence and Mutual Exclusivity (b)

Independence: Two events A and B are independent if \( P(A ∩ B) = P(A) ⋅ P(B) \) . Here, A is 'female' and B is 'prefers watching sports' .If 'female' and 'watching sports' are independent, \( P(F ∩ S) = P(F) ⋅ P(S) \)= 0.52 ⋅ 0.564 = 0.293. This is not equal to P(F ∩ S) = 45 / 250 = 0.18, so 'female' and 'watching sports' are not independent. Mutual Exclusivity: Events are mutually exclusive if they cannot occur at the same time. In this case, a person can be a 'female' and 'prefer watching sports' at the same time, so 'female' and 'watching sports' are not mutually exclusive.

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Key Concepts

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

Two-way table

A two-way table is a helpful tool that organizes data into categories based on two different variables. In the example provided, we examine adults who either prefer watching sports or opera on television, categorized by gender - "Male" or "Female." This table helps in visualizing relationships between these two variables.

• The vertical columns represent the preference, i.e., sports or opera.
• The horizontal rows indicate gender, i.e., male or female.

This layout enables easy calculation of totals and ensure that each category intersection is clear, like how many males prefer opera or females prefer sports. By using a two-way table, we can quickly determine probabilities or detect associations between different categories.
Understanding these associations is crucial when we move into more complex statistics like conditional probabilities and independence of events.

Independence of events

Two events are said to be independent if the occurrence of one does not affect the probability of the other. We check independence using the formula: \[ P(A \cap B) = P(A) \cdot P(B) \].
In the context of our exercise, events like "Being female" (denoted as Event A) and "Preferring sports" (Event B) were checked for independence.
Here, event A and B are dependent if their joint probability \( P(F \cap S) \) is not equal to the product of their individual probabilities \( P(F) \) and \( P(S) \).
In the exercise:

• Total number of females = 130.
• Total number preferring sports = 141.
• Joint occurrence (Female and sports preference) = 45.
• Calculated \( P(F \cap S) = 45/250 \) which is 0.18.
• Calculated \( P(F) \cdot P(S) = 0.52 \cdot 0.564 = 0.293 \).

Since these probabilities are not equal, the events "Being female" and "Preferring sports" are not independent.

Mutually exclusive

Mutually exclusive events cannot happen at the same time. If one event occurs, the other cannot. For example, drawing an Ace and a King from a standard deck of cards in a single draw are mutually exclusive events.
In our exercise, the query was whether "Being Female" and "Preferring Sports" could be mutually exclusive.
However, as we see in the table:

• There are 45 females who prefer sports.

This means these events can occur simultaneously, as people who are female can prefer watching sports.
Thus, "Female" and "Preferring Sports" are not mutually exclusive events. They can and do occur together, illustrated by the existence of females who enjoy watching sports, directly contradicting the definition of mutual exclusivity.

Conditional probability

Conditional probability measures the likelihood of an event occurring, given that another event has already occurred. It is denoted as \( P(A|B) \), meaning the probability of event A occurring given that B has occurred.
In the exercise:

• For the probability of a female preferring sports \( P(S|F) \) is calculated as \( \text{Females who prefer sports} / \text{Total females} \).
• Given values from the table: 45 females prefer sports, out of 130 total females. \( P(S|F) = 45/130 = 0.346 \).
• Similarly, the probability of being male given sports preference \( P(M|S) \) is \( \text{Males who prefer sports} / \text{Total sports preferring individuals} \) which is \( 96/141 = 0.681 \).

Understanding conditional probability allows us to evaluate scenarios where one condition directly influences the likelihood of another event, providing insightful context into real-world decisions, such as targeting a marketing audience by preference.