Two dice are thrown simultaneously: Find the probability of its occurrence is
An even number as the sum
The sum as a prime number
A total of at least 10
A doublet of even number
A multiple of 2 on one dice and a multiple of 3 on the other dice.
Same number on both dice
A multiple of 3 as the sum

Explanation

Sample space for total number of possible outcomes
`(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)`
Total number of outcomes `=36`

(i)Favorable outcomes for sum as prime are
`(1,1),(1,2),(1,4),(1,6),(2,3),(2,3),(2,5),(3,2),(3,4),(4,1),(4,3),(5,2),(5,6),(6,1),(6,5)`
Number of favorable outcomes `=15`
Hence, the probability of getting the sum as a prime number `=15/36=5/12`
(ii)Favorable outcomes for total of atleast `10` are`(4,6),(5,5),(5,6),(6,4),(6,5),(6,6)`
Number of favorable outcomes `=6`
Hence, the probability of getting a total of atleast `10= 6/36=1/6`

(iii)Favorable outcomes for a doublet of even number are
`(2,2),(4,4),(6,6)`
Number of favorable outcomes `=3`
Hence, the probability of getting a doublet of even number `=3/36=1/12`

(iv)Favorable outcomes for a multiple of `2` on one dice and a multiple of `3` on the other dice are
`(2,3),(2,6),(3,2),(3,4),(3,6),(4,3),(4,6),(6,2),(6,3),(6,4),(6,6)`
Number of favorable outcomes `=11`
Hence, the probability of getting a multiple of `2` on one dice and a multiple of `3` on the other dice `=11/36`​

(v)Favorable outcomes for getting a multiple of `3` as the sum
`(1,2),(1,5),(2,1),(2,4),(3,3),(3,6),(4,2),(4,5),(5,1),(5,4),(6,3)(6,6)`
Number of favorable outcomes `=12`
Hence, the probability of getting a multiple of `3` as the sum `=12/36​=1/3`