Question

Roll a die (a) What is the probability of getting the first 6 at or before the fifth roll? (b) What is the probability of getting the third 6 at the tenth roll? (c) What is the expected number of rolls to get the fifth 6? (d) Given that the second 6 occurred at the tenth roll, what is the probability that the first 6 occurred at the fifth roll?

Answer 1

probability of a 6 on a roll p=1/6

a) probability of getting the first 6 at or before the fifth roll=1-P(no 6 tll first 5 rolls)=1-(1-1/6)⁵ =0.5981

b)

probability of getting the third 6 at the tenth roll=P(exact two 6 till 9 roll and 6 on 10th roll)

= =0.0465

c)

expected number of rolls to get the fifth 6 =r/p=5/(1/6)=30

d)

probability that the first 6 occurred at the fifth roll given second 6 occurred at the tenth roll

=P(first 6 on 5th roll and seconf 6 on 10th roll)/P(2nd 6 on 10th roll)

=(5/6)⁴*(1/6)*(5/6)⁴*(1/6)/ =0.1111