Let P a random variable which is uniformly distributed betwe

Let P a random variable which is uniformly distributed between 0 and 1. On any given day, a particular machine is functional with probability P. Furthermore, given the value of P, the status of the machine on different days is independent

1)Find the probability that the machine is functional on a particular day.

2)Let A be the event that the machine was functional on m out of the last n days. Find the conditional PDF of P given A. You may use the identity?₀¹ p^k(1-p)^n-kdp=k!(n-k)!/(n+1)!.
f_P|A(0.7|m=2,n=3)=

3)Find the conditional probability that the machine is functional today given that it was functional on 5 out of the last 6 days.

Solution

1) Prob that machine funcitons on any day = p

2) P(A) = nCmp^m(1-p)^n-m

f(P/A) = f(PA)/f(A)