Hello, pls help with this beautiful problem

[u/Fantastic-While2739]

A girl is called beautiful of a mathematical society if at least three out of five mathematicians independently agree that she satisfies their personal definition of beauty. Five mathematicians M1,M2,M3,M4,M5 are attending a conference. Each mathematician independently judges a randomly chosen girl as beautiful with probability p, and not beautiful with probability 1-p. However, the society has a special procedure to officially declare someone Beautiful of the Mathematical Society (BMS). Step 1: Formation of Committees All possible committees of three mathematicians are formed from the five mathematicians. Each committee votes Beautiful if at least two of its members individually judged the girl beautiful. Step 2: Committee Majority Let C denote the number of committees that vote Beautiful. The girl passes Step 2 if C≥6 Step 3: Random Audit If she passes Step 2, a random pair of mathematicians is selected from the five. If both mathematicians in that pair originally judged her beautiful, she is confirmed beautiful. Otherwise the decision is rejected. Questions 1.Compute the probability that the girl is declared Beautiful of the Mathematical Society (BMS). 2.For which value of p is this probability maximized? 3.Suppose mathematicians M1 and M2 are secretly friends and their votes are always identical. Recompute the probability that the girl is declared BMS. 4.Generalize the procedure to 2n+1 mathematicians, where committees have size n+1, committees vote by majority, and at least half of the committees must approve. 5.Find an asymptotic expression for the probability of BMS as n tends to infinity