If two dice are rolled simultaneously, find the probability of the following events:
(1) The sum of the digits on the upper face is at least 10.
(2) The sum of the digits on the upper face is 33.
(3) The digit on the first die is greater than the digit on the second die.

Answer

(1) `(1)/(6)`
(2) 0
(3) `(5)/(12)`

Explanation

Two dice are rolled simultaneously.
`therefore"the sample space"`
`S={(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)}`
`thereforen(S)=36.`
(1) "Let A be the event that the sum of the digits on the upper faces is at least 10."
Then A = `{(4,6),(5,5),(5,6),(6,4),(6,5),(6,6)}`
`thereforen(A)=6.`
`P(A)=(n(A))/(n(S))=(6)/(36)=(1)/(6).`
(2) Let B be the event that the sum of the digits on the upper face is 33.
The sum of the digits on the upper face is maximum (6,6) that is 12.
`thereforeB={}." "thereforen(B)=0`
`P(B)=(n(B))/(n(S))=(0)/(36)=0.`
Let C be the event that the digit on the first die is greater than the digit on the first die is greater than the digit on the second die.
Then`C={(2,1),(3,1),(3,2),(4,1),(4,2),(4,3),`
`(5,1),(5,2),(5,3),(5,4),(6,1),(6,2),(6,3),`
`(6,4),(6,5)}`
`thereforen(C)=15.`
`P(C)=(n(C))/(n(S))=(15)/(36)=(5)/(12).`