Question

I want to see the thought process behind this so that I could approach similar problems on my own, so please attempt to show every step and formula used, thank you.

9. Two fair six sided dice are rolled (i) What is the probability that the sum of the two results is 6? (ii) What is the probability that the largest of the two results is 4? (ii) What is the probability that the both results are at most 4? (iv) What is the probability that the number 3 appears at least once?

Answer 1

(9)

(1) The question asks that what is the probability that the sum of the two results is 6?

So, we will find such pairs whose sum results 6.

The pairs are:-

(1,5) (5,1) , (2,4) (4,2), (3,3) = 5 pairs or outcomes

Their are no more pairs other than these whose sum is 6

And we know that total no. Of possible outcomes are = (6)² = 36

So, the probability that the sum of the two results 6 = 5/36

(2) The pairs according to question are:-, one of the two dices should have 4 and the other not more than 4

(1,4) , (2,4) , (3,4) , (4,4) ( 4,1), (4,2) , (4,3) = 7 pairs or outcomes

Total no. Of outcomes = 36

Probability that the largest of the two results is 4 = 7/36

(3) The pairs according to question are:- , both the dices should not exceed 4

(1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4) = 16 pairs or outcomes

Total no. Of outcomes = 36

Probability that both the results are at most 4 = 16/36 = 4/9

(4) The pairs according to question are:-, no. 3 should appear at least once, it can appear twice also

(1,3), (2,3), (3,3), (4,3), (5,3) (6,3), (3,1), (3,2), (3,4), (3,5), (3,6) = 11 pairs or outcomes

Total no. Of outcomes = 36

Probability that the no. 3 appears atleast once = 11/36