A black and a red dice are rolled. (a)   Find the conditional probability of obtaining a sum greater than 9. Given that the black die resulted in a 5. (b)   Find the conditional probability of obtaining the sum 8? given that the red die resulted in a number less than 4.

Explanation

It is on a condition that a black and a red dice is rolled.
When `2` dices are thrown simultaneously,
the total range of outcome is `36`,
so the nrange of sample space or possible outcome is `36`.
(a)It is on condition that a black dice is resulted in a five .
Let A be the event that total of the amount on dices is larger than nine .
`A={ ( 4,6 ),( 5,5 ),( 5,6 ),( 6,4 ),( 6,5 ),( 6,6 ) }`
so,`P(A)=6/36`
`=1/6`
Let B be the event that black dice results in 5.


`B={ ( 5,1 ),( 5,2 ),( 5,3 ),( 5,4 ),( 5,5 ),( 5,6 ) }`
`P(B)=6/36`
`=1/6`
The common outcomes of the event A and B are,
`A∩B={ ( 5,5 ),( 5,6 ) }`
`P(A∩B)=2/36`
`=1/18`
The conditional probability of the given event is given by `P( A/B )`
`=(P(A∩B))/(P(B))=(1/18)/(1/6)`
`=6/18=1/3`
Therefore, the conditional probability of obtaining a sum greater than `9`, given that black dice is resulted in` 5`, is `1/3`

(b)It is given that the red dice is resulted in a number less than `4`.


Let E be the event that sum of the number on dices is `8`.


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