We have two black boxes I and II Box I contains four balls n

We have two black boxes: I and II. Box I contains four balls numbered 1,2,3,4, and box II - three balls numbered 4,5,6. The following experiment is conducted. A biased coin is flipped. If the outcome is H, then a ball is drawn randomly from box I, otherwise - a ball is drawn randomly from box II. Let X be the number of the drawn ball in the end of the experiment. Given that X = 4, what is the probability that box I was chosen at the first step (i.e., the coin showed H)? Define events A = {coin shows H} and B = {X is even}. For which values of p (recall 0

Solution

(a)P(H|X=4)=(p/4 + (1-p)/3)/(1/4 + 1/3) =(4-p)/7

(b)

P(AB)=(1/2)*(p)

P(A)=p

P(B)=p/2 +2(1-p)/3=(4-p)/6

For independence,

P(AB)=P(A)P(B)

=>p/2=(4-p)p/6

=>p²-p=0

=>p=0,1

(c)E(X)=(1+2+3+4)*(p/4)+(4+5+6)*(1-p)/3=5p/2 + 5(1-p)=5(2p-1)/2

(d)P(X=x)=p/4 if x=1,2,3,4

=(1-p)/2 if x=5,6

It is uniform distribution if-

p/4 =(1-p)/2

=>p=2/3