What is the probability that at least one of a pair of fair 6-sided dice shows the number 2, given that the sum of the dice is 4?

Question

HINT: Write out all the possible ways to get a sum of 4 on a pair of 6-sided dice, assuming that the order of the dice matters.

Answer 1

The sum of the dice is four : (1,3), (2,2), (3,1) => P(B) = 3/36 {Denominator :- pair of dice= 6^2}

At least one of a pair shows two : (2,2) => P(A intersection B) = 1/36

The question says given that sum of the dice is 4, so when the sum occurred first(denominator) what is the probability that 2 is showed(numerator).

The condition P(A intersection B) is occurred after P(B) is occurred. Hence, it is conditional.

According to conditional probability : P(A|B) = P (A intersection B) / P(B) => (1/36) / (3/36) = 1/3

Answer 2

There are 3 ways for the sum of the dice to show up as 4:

1 on die 1 and 3 on die 2

3 on die 1 and 1 on die 2

2 on die 1 and 2 on die 2

So the probability that one of the die show a 2 is 1/3.