Understanding of the full probability formula and Bayesian Formula

Understanding of the full probability formula and Bayesian Formula

How can I understand these two formulas? For example, if you apply for a scholarship from a school, you can only get this scholarship if you meet certain conditions. So what are the reasons for making it possible for you to receive a scholarship? 1. The probability of a scholarship is P (A1) = 0.3. 2. 4. The probability of a scholarship is P (A2) = 0.4. 3. For five top students, the probability of a scholarship is P (A3) = 0.5. 4. 6. The probability of obtaining a scholarship is P (A4) = 0.6. These students are only one of the three good, four good, five good, and six good students. In this school, the probability of being a good student is P (B1) = 0.4, the probability of being a good student is P (B2) = 0.3, and the probability of being a good student is P (B3) = 0.2, 6. The probability is P (B4) = 0.1. Now the question is, what is the probability that a student will receive a scholarship?

What are the ways for a student to receive a scholarship? This student is a good student. He just applied for the scholarship p1 = P (A1) * P (b1 | A1) = 0.4*0.3 = 0.12; this student is a four-student. He just got the scholarship as a four-student. P2 = P (A2) * P (B2 | A2) = 0.3*0.4 = 0.12; this student is a five-year-old student and has just received a scholarship as a five-year-old student. P3 = P (A3) * P (B3 | A3) = 0.2*0.5 = 0.10; this student is a good six student. He just got the scholarship as a good six student. P4 = P (A4) * P (B4 | A4) = 0.1*0.6 = 0.06. If one student can receive a scholarship in either of the four ways, the probability of obtaining the scholarship is P = p1 + p2 + P3 + P4 = 0. 4. therefore, we can understand the full probability formula as follows: there are many causes of an event (mutually exclusive for various reasons). The probability of this event is the sum of the probability of this event occurring for each reason.

A student has received a scholarship. What is the probability of being a good student? P = p1/(P1 + p2 + P3 + P4) = 0.3. How can this problem be solved? An event has already occurred, which can be caused by many reasons. What is the probability of this event as a result of one of the reasons? This is the problem solved by Bayesian probability formula. Just as a book has now been borrowed by someone else (the event has occurred), it is known that only three people, Zhang San, Li Si, and Wang Wu, are borrowed (all the reasons for the event ). So what is the probability that this book will be lent by James?

Have you understood these two formulas.

 

Understanding of the full probability formula and Bayesian Formula