- The probability of being blond is \(P(E) = \frac{60}{100} = 60\%\).
- The conditional probability of being blond given being a junior is \(P(E \mid F) = \frac{P(E \cap F)}{P(F)} = \frac{0.30}{0.50} = 0.60\).
- Since \(P(E \mid F) = P(E)\), the events are independent.
- \(P(E \mid F) = \boxed{0.6}\)
- \(P(E) = \boxed{60}\%\)
- Events \(E\) and \(F\) are \(\boxed{\text{independent}}\).
Explanation
- Analyze the problem and identify data
The problem asks us to determine if two events—being a blond student (\$E\$) and being a junior student (\$F\$)—are independent, based on the provided data in a two-way table. We start by gathering the relevant probabilities directly from the table. The table shows the distribution of students by grade (Sophomore, Junior, Senior) and hair color (Blond, Brown) in percentages. The total number of students is represented by 100%, so we can treat these percentages directly as probabilities.
- Extract probabilities from the table
First, we identify the probabilities of our events of interest:
- Event \$E\$ is the event that a student is blond. Looking at the "Total" row for the "Blond" category, we see that 60% of all students are blond. Therefore, \(P(E) = \frac{60}{100} = 0.60\), or 60%.
- Event \$F\$ is the event that a student is a junior. Looking at the "Total" column for the "Jr" grade, we see that 50% of all students are juniors. Therefore, \(P(F) = \frac{50}{100} = 0.50\), or 50%.
- The intersection, \(E \cap F\), represents students who are both blond and juniors. Looking at the intersection of the "Blond" row and "Jr" column, we find 30%. Therefore, \(P(E \cap F) = \frac{30}{100} = 0.30\).
- Calculate conditional probability \(P(E \mid F)\)
To determine if events \$E\$ and \$F\$ are independent, we use the definition of conditional probability. The probability of a student being blond given that they are a junior is calculated as:
\[P(E \mid F) = \frac{P(E \cap F)}{P(F)}\]
Substituting the values we found:
\[P(E \mid F) = \frac{0.30}{0.50} = 0.60\]
So, the probability that a student is blond given that they are a junior is 0.60, or 60%.
- Check for independence and conclude
Now, we compare the conditional probability \(P(E \mid F)\) to the marginal probability \(P(E)\).
- We calculated \(P(E \mid F) = 0.60\).
- From our initial identification, \(P(E) = 0.60\).
Since \(P(E \mid F) = P(E)\), the occurrence of event \$F\$ does not change the probability of event \$E\$ occurring. This is the definition of independence. Therefore, the events are independent.
Examples
This concept is used in market research to determine if two customer demographics are related. For instance, a store might check if a customer's age group is independent of their preference for a specific product category to decide how to target their advertisements effectively. If independent, they can use general marketing strategies; if dependent, they must tailor their approach to specific demographics.