This table shows how many students from two high schools attended a football game.
\begin{tabular}{|l|c|c|c|}
\hline & Attended the game & \begin{tabular}{c}
Did not attend
the game
\end{tabular} & Total \\
\hline Westville & 60 & 90 & 150 \\
\hline North Beach & 90 & 110 & 200 \\
\hline Total & 150 & 200 & 350 \\
\hline
\end{tabular}
A student is randomly selected. What is the probability that a student did not attend the game, given that the student is from Westville? Round your answer to two decimal places.
A. 0.45
B. 0.57
C. 0.43
D. 0.60
Answer (2)
Define events for attending/not attending the game and being from Westville.
Apply the conditional probability formula: P ( N A ∣ W ) = P ( W ) P ( N A ∩ W ) .
Extract the necessary probabilities from the table: P ( N A ∩ W ) = 350 90 and P ( W ) = 350 150 .
Calculate the conditional probability: P ( N A ∣ W ) = 150 90 = 0.6 . The final answer is 0.60 .
Explanation
Understand the problem We are given a table that shows the number of students from Westville and North Beach high schools who attended or did not attend a football game. We want to find the probability that a randomly selected student did not attend the game, given that the student is from Westville. This is a conditional probability problem.
Define events Let A be the event that a student attended the game. Let N A be the event that a student did not attend the game. Let W be the event that a student is from Westville. Let NB be the event that a student is from North Beach. We want to find P ( N A ∣ W ) , which is the probability that a student did not attend the game, given that the student is from Westville.
Apply conditional probability formula Using the conditional probability formula, we have: P ( N A ∣ W ) = P ( W ) P ( N A ∩ W )
Extract data from the table From the table, we can see that: Number of students from Westville who did not attend the game = 90 Total number of students from Westville = 150 Total number of students = 350
Calculate the conditional probability Therefore: P ( N A ∩ W ) = 350 90 P ( W ) = 350 150 So, P ( N A ∣ W ) = 350 150 350 90 = 150 90 = 5 3 = 0.6
State the final answer The probability that a student did not attend the game, given that the student is from Westville is 0.6. Rounding to two decimal places, we get 0.60.
Examples
Conditional probability is used in many real-world scenarios, such as medical diagnosis, risk assessment, and marketing. For example, a doctor might use conditional probability to determine the probability that a patient has a certain disease, given that the patient has certain symptoms. In marketing, conditional probability can be used to determine the probability that a customer will buy a product, given that the customer has certain demographics or purchase history. Understanding conditional probability helps in making informed decisions based on available data.
The probability that a student did not attend the game, given that the student is from Westville, is calculated to be 0.6. After rounding to two decimal places, the final answer is 0.60. Therefore, the correct option is D: 0.60.
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