Question

A player throws an unbiased dice, whenever his throw results in a score 1 or 2, he gets an additional throw. Then, the probability to get the sum of score(s) = n is

• A. o if n=1
• B. $\frac{49}{6^3}$ if n=5
  
• C. $\frac{307}{6^4}$ if n=6
  
• D. $\frac{517}{6^5}$ if n=7

Answer 1

Answer: (A), (B), (C)

The correct options are
A o if n=1
B $\frac{49}{6^3}$ if n=5

C $\frac{307}{6^4}$ if n=6


Score of 5 can be obtained as
5,(1,4), (1,1,3), (2,3)

P(scoring 5)=$\frac{1}{6}+\frac{1}{6^2}+\frac{1}{6^2}+\frac{1}{6^3}=\frac{49}{216}$

Score of 1 can not be obtained
So P(scoring 1)=0

Score of 6 can be obtained as
6,(1,5), (2,4), (1,1,4), (1,1,1,3),(1,2,3), (2,1,3),

P(scoring) =$\frac{1}{6}+\frac{2}{6^2}+\frac{3}{6^3}+\frac{1}{6^4}=\frac{307}{1296}$

Score of 7 can be obtained as
(1,6), (1,1,5), (2,5), (1,1,1,4)(1,2,4), (2,1,4),
(1,1,1,1,3), (1,1,2,3), (2,1,1,3), (1,2,1,3), (2,2,3)

P (scoring 7) $=\frac{2}{6^2}+\frac{4}{6^3}+\frac{4}{6^4}+\frac{1}{6^5}=\frac{601}{65}$