Final answer:
The probability that an entering freshman with a good high-school GPA and high SAT scores will graduate with a BA is a range of values, not a specific number. It depends on the prior probability of getting a BA given a good high school GPA.
Step-by-step explanation:
To find the probability that an entering freshman with a good high-school GPA and high SAT scores will graduate with a BA, we need to use conditional probability. Let's denote the event that a student has a good high school GPA as 'G', the event that a student has high SAT scores as 'S', and the event that a student graduates with a BA as 'B'. We are given that P(B|G) = 0.90 (the probability of getting a BA given a good high school GPA) and P(B'|G') = 0.55 (the probability of not getting a BA given not having a good high school GPA). We also know that P(S|B) = 0.90 (the probability of having high SAT scores given a BA). Using Bayes' theorem, we can calculate the probability as follows:
P(B|G&S) = P(G&S|B) * P(B) / P(G&S)
P(G&S|B) = P(S|B) * P(G|B) = 0.90 * P(B|G) = 0.90 * 0.90 = 0.81
P(B) = P(B|G) * P(G) + P(B|G') * P(G') = 0.90 * P(G) + 0.55 * (1 - P(G)) = 0.90 * P(G) + 0.55 - 0.55 * P(G) = 0.55 + 0.35 * P(G)
P(G&S) = P(S|G) * P(G) = 0.90 * P(G)
Substituting these values into the equation above:
P(B|G&S) = 0.81 * (0.55 + 0.35 * P(G)) / (0.90 * P(G)) = 0.567 + 0.063 * P(G)
Since P(G) is the prior probability that an entering undergraduate student will get their BA if they have a good high school GPA, it is a value between 0 and 1. Therefore, the probability that an entering freshman with a good high-school GPA and high SAT scores will graduate with a BA is a range of values, not a specific number. The probability increases as P(G) increases, but to find the precise probability, we would need to know the exact value of P(G).