Question

Suppose you have a die that has probability p of resulting in the outcome 6 when rolled, where p is a continuous random variable that is uniformly distributed over [0,1/3]. Suppose you start rolling this die. (The value of p does not change once you start rolling.) Give exact answers as simplified fractions.

a) Compute the probability that the first roll is 6.

b) Compute the probability that the first two rolls are both 6.

c) Let Sy be the event that the first roll is 6 and S2 be the event that the second roll is 6. Compute P (S2 Si).

Answer 1

pq no 9 Answer: Given that, prior PDF of p is Pp (P)= 3;0&p=1/3 a) the conditional Pyobability that the first roll is The unconditional probability is PlFirst Yo!! is 6) = ( pfp(p) dp 113 - Podp my alco - B x 3 1: P[first roll is 6) = 116] b) The conditional probability that the first two rolls are both 6 is Pe first two 10113 are both 6lpapp

Pg No@ The unconditional probabi P(fryst keo rollo ave both 6) = s PPRC Pop P[first two vols are box, 6) = s'® playop . [per la L.:. PlFirst two rolls are both 6): 11271 since, the 2015 are Independent PC sa 151; P) = P2 52; P) =P Pl52151)== -> [from partlay] . Pesels) =