Question

Problem:

Roll a pair of fair dice. Find the probability that:

a) On an individual roll, the sum is 5 or 11.

b) The first time that you roll a sum of 5 or 11 is on your 7th roll.

c) The 3rd time you roll a sum of 5 or 5 is on your 7th roll.

Solutions from back of textbook:

a) 0.1667

b) 5.58%

c) 3.35%

Answer 1

a)

total number of outcome on two dice =6*6 =36

number of ways sum can be 5 =4 ((1,4),(2,3),(3,2),(4,1))

number of ways sum can be 11 =2 ((6,5),(5,6))

P( sum is 5 or 11)=(4+2)/36 =1/6 =0.1667

b)

The first time that you roll a sum of 5 or 11 is on your 7th roll =P(no 5 or 7 in first 6 rolls and on 7th roll first 5 or 11) =(1-0.1667)^6*0.1667=0.0558 ~ 5.58%

c)P( 3rd time you roll a sum of 5 or 5 is on your 7th roll ) =P(exactly 2 5 or 11 in first 6 rolls and 5 or 11 on 7th roll) =(₆C₂)*(0.1667)²*(1-0.1667)⁴*0.1667 =0.2010*0.1667 =0.0335 ~3.35%