Two dice are rolled [probability]

Question

A) Find the probability that the first number is 3 given that the sum is 7.

 

B) Find the probability that the first number is 3 given that the sum is 8.

 

I'm not understanding how the answers are A) 1/6 and B) 1/5

Answer 1

When you roll a single die there are 6 equally likely possible outcomes:  1,2,3,4,5 or 6  

 

When you roll 2 dice there are 36 possible outcomes that can be represented by ordered pairs (X,Y) where each of X and Y are 1 of the 6 possible outcomes listed above.  These outcomes are  

 

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)

(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)

(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)

(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)

(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)

(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)  

 

There are 6 outcomes that result in a sum of 7:  they are  

 

(1,6)  (2,5)  (3,4)  (4,3)  (5,2)  (6,1)  

 

Of these 6 outcomes, only 1 [the (3,4) outcome] has the first number equal to 3.  So if the sum is 7, the probability that the first number is a 3 is 1/6  

 

There are 5 outcomes that result in a sum of 8:  they are  

 

(2,6)  (3,5)  (4,4)  (5,3)  (6,2)  

 

Of these 5 outcomes, only 1 [the (3,5) outcome] has the first number equal to 3.  So if the sum is 8, the probability that the first number is a 3 is 1/5