Answer :
To find the probability that the student actually has a SAT score above 1500, given that the College Board reports a score above 700, we can use a tree diagram. The probability is approximately 0.727.
Let A be the event that the student has a score above 1500, and let B be the event that the College Board reports a score above 700. We want to find P(A|B), the probability that the student has a score above 1500 given that the College Board reports a score above 700.
The probability that a student in the general population has a score above 1500 is 0.10, so P(A) = 0.10.
The College Board can give a correct report 96% of the time, so P(B|A) = 0.96, and it can also give a correct report 96% of the time for a student with a score below 1500, so P(B|A') = 0.96.
Using the formula for conditional probability, we have:
P(A|B) = P(A and B) / P(B) = P(B|A) * P(A) / P(B|A) * P(A) + P(B|A') * P(A') = 0.96 * 0.10 / (0.96 * 0.10 + 0.04 * 0.90) = 0.96 * 0.10 / (0.096 + 0.036) = 0.096 / 0.132 = 0.727
Therefore, the probability that the student actually has a SAT score above 1500, given that the College Board reports a score above 700, is approximately 0.727.
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Final answer:
The probability that the student actually has a SAT score above 1500, given that the College Board reports a score above 700, is approximately 0.96.
Explanation:
To solve this problem, we can construct a tree diagram to visualize the different scenarios and calculate the desired probability.
Let's denote the event that the student has a SAT score above 1500 as A, and the event that the College Board reports a score above 700 as B.
Based on the given information, we know that the probability of event A, P(A), is 0.10 (10% of students have a SAT score above 1500), and the probability of event B given A, P(B|A), is 0.96 (the College Board can give a correct report 96% of the time).
Using the tree diagram, we can calculate the probability of event A given B, P(A|B), which is the probability that the student actually has a SAT score above 1500, given that the College Board reports a score above 700.
Let's construct the tree diagram:
From the tree diagram, we can see that there are two possible outcomes: the student has a SAT score above 1500 and the College Board reports a score above 700 (event AB), and the student has a SAT score below 1500 and the College Board reports a score above 700 (event ¬A¬B).
The probability of event AB, P(AB), is calculated as the product of the probabilities of event A and event B given A: P(AB) = P(A) * P(B|A) = 0.10 * 0.96 = 0.096.
The probability of event ¬A¬B, P(¬A¬B), is calculated as the product of the probabilities of event ¬A and event ¬B given ¬A: P(¬A¬B) = P(¬A) * P(¬B|¬A) = (1 - P(A)) * (1 - P(B|¬A)) = (1 - 0.10) * (1 - 0.96) = 0.004.
The sum of these probabilities gives us the total probability of event B, P(B): P(B) = P(AB) + P(¬A¬B) = 0.096 + 0.004 = 0.1.
Finally, we can calculate the probability of event A given B, P(A|B), using the formula for conditional probability: P(A|B) = P(AB) / P(B) = 0.096 / 0.1 = 0.96.
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