A student has enrolled for the first time at a University. The probability of receiving it is 85% if you obtain a scholarship and 45% if you do not obtain one. If the probability of obtaining a scholarship is 30%. i) Determine the probability that the student receives a scholarship. ii) If the student receives a scholarship, determine the probability that he or she received a scholarship.

To solve this problem, we can use conditional probability. Let's break it down step by step: i) Determine the probability that the student receives a scholarship. The probability of receiving a scholarship is given as 30%. Therefore, P(Scholarship) = 0.30 or 30%. ii) If the student receives a scholarship, determine the probability that he or she received a scholarship. This is a conditional probability problem, and we want to find the probability that the student received a scholarship given that they received it. We can denote this as P(Scholarship | Received Scholarship). We can use Bayes' theorem to calculate this: P(Scholarship | Received Scholarship) = [P(Received Scholarship | Scholarship) * P(Scholarship)] / P(Received Scholarship) We already know: - P(Scholarship) = 0.30 (from part i) - P(Received Scholarship | Scholarship) = 0.85 (probability of receiving if you obtain a scholarship) Now, we need to find P(Received Scholarship), which can be calculated using the law of total probability. We can calculate it as follows: P(Received Scholarship) = P(Received Scholarship | Scholarship) * P(Scholarship) + P(Received Scholarship | No Scholarship) * P(No Scholarship) - P(Received Scholarship | No Scholarship) = 0.45 (probability of receiving if you do not obtain a scholarship) - P(No Scholarship) = 1 - P(Scholarship) = 1 - 0.30 = 0.70 (probability of not obtaining a scholarship) Now we can calculate P(Received Scholarship): P(Received Scholarship) = (0.85 * 0.30) + (0.45 * 0.70) P(Received Scholarship) = 0.255 + 0.315 P(Received Scholarship) = 0.57 Now, we can calculate P(Scholarship | Received Scholarship) using Bayes' theorem: P(Scholarship | Received Scholarship) = (0.85 * 0.30) / 0.57 P(Scholarship | Received Scholarship) = 0.255 / 0.57 P(Scholarship | Received Scholarship) ≈ 0.4474 So, if the student receives a scholarship, the probability that they received a scholarship is approximately 0.4474, or 44.74%.