Question

Suppose that we roll a pair of (6 sided) dice until the first sum value appears that is 7 or less, and then we stop afterwards.

a. What is the probability that exactly three (pairs of) rolls are required?

b. What is the probability that at least three (pairs of) rolls are needed?

c. What is the probability that, on the last rolled pair, we get a result of exactly 7?

Answer 1

here P(7 or less on a roll of dice)=21/36

and P(more than 7 on a roll of dice)=15/36

a)

probability that exactly three (pairs of) rolls are required =P(first 2 roll show more than 7)*P(3rd roll shows 7 or less than that)

=(15/36)²*(21/36)=0.1013

b)

probability that at least three (pairs of) rolls are needed=P(first 2 rolls show a number greater than 7)

=(15/36)² =0.1736

c)

P(on the last rolled pair, we get a result of exactly 7)=P(first roll shows 7)+P(first roll more than 7 and second shows exactly 7)+P(first 2 roll more than 7 and second shows exactly 7)+........

=(6/36)+(15/36)*(6/36)+(15/36)²*(6/36)+(15/36)³*(6/36)+...........

=(6/36)/(1-15/36)=6/21=2/7