This table shows how many sophomores and juniors attended two school events. A student is selected randomly from this group.
\begin{tabular}{|c|c|c|c|}
\hline & \begin{tabular}{c}
Jazz band \
concert
\end{tabular} & Volleyball game & Total \
\hline Sophomore & 35 & 42 & 77 \
\hline Junior & 36 & 24 & 60 \
\hline Total & 71 & 66 & 137 \
\hline
\end{tabular}
What is the probability that the student attended the volleyball game, given that the student is a sophomore?
Round your answer to two decimal places.
A. 0.18
B. 0.55
C. 0.45
D. 0.36
Answer (2)
Define events: S (sophomore), V (attended volleyball).
Calculate conditional probability: P ( V ∣ S ) = P ( S ) P ( V S ) .
Find probabilities from table: P ( V S ) = 137 42 , P ( S ) = 137 77 .
Calculate and round: P ( V ∣ S ) = 77 42 ≈ 0.55 .
Explanation
Understand the problem We are given a table that shows the number of sophomores and juniors who attended two school events: a Jazz band concert and a Volleyball game. We want to find the probability that a randomly selected student attended the volleyball game, given that the student is a sophomore.
Define events and conditional probability formula Let S be the event that the student is a sophomore, and let V be the event that the student attended the volleyball game. We want to find the conditional probability P ( V ∣ S ) , which is the probability that the student attended the volleyball game given that the student is a sophomore. The formula for conditional probability is: P ( V ∣ S ) = P ( S ) P ( V S )
Find probabilities from the table From the table, we can see that there are 42 sophomores who attended the volleyball game. The total number of sophomores is 77. Therefore, the probability that a randomly selected student is a sophomore and attended the volleyball game is: P ( V S ) = 137 42 The probability that a randomly selected student is a sophomore is: P ( S ) = 137 77
Calculate the conditional probability Now we can calculate the conditional probability: P ( V ∣ S ) = P ( S ) P ( V S ) = 137 77 137 42 = 77 42 We can simplify the fraction by dividing both the numerator and the denominator by 7: 77 42 = 777 427 = 11 6
Convert to decimal and round To round the answer to two decimal places, we can divide 6 by 11: 11 6 = 0.545454... Rounding to two decimal places, we get 0.55.
State the final answer The probability that the student attended the volleyball game, given that the student is a sophomore, is approximately 0.55.
Examples
This type of probability calculation is used in marketing to understand customer behavior. For example, if a company knows that 60% of their customers are women and 20% of their customers are women who buy a specific product, they can calculate the probability that a customer who buys the product is a woman. This helps them target their marketing efforts more effectively.
The conditional probability that a student attended the volleyball game given they are a sophomore is calculated to be approximately 0.55. This means there is a 55% chance that a randomly selected sophomore attended the volleyball game. Thus, the answer is option B.
;
