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.

Answer

N/a

Explanation

Let first the black dice and thean a red dice is thrown. So the number of elements in the sample space `=6xx6=36`
(i) Let `E=` event of getting a sum of greater than 9.
and `F=` event of getting 5 on black dice.
Therefore, `E={(6,4),(4,6),(5,5),(5,6),(6,5),(6,6)}`
`implies n(E)=6`
and `F={(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)}`
`impliesn(F)=6`
`impliesEnnF={(5,5),(5,6)impliesn(EnnF)}=2`
`P(E)=6/36=1/6`
`P(F)=6/36=1/6` and `P(EnnF)=2/36=1/18`
`:.P(E/F)=(P(EnnF))/(P(F))=(1//18)/(1//6)=1/18xx6/1=1/3`
(ii) Let `E=` event of getting a sum of 8.
and `F=` event of getting a number less than 4.
`:.E={(2,6),(3,5),(4,4),(5,3),(6,2)}`
`impliesn(E)=5`
and
`F={{:((1,1)","(1,2)","(1,3)","(2,1)","(2,2)","(2,3)),((3,1)","(3,2)","(3,3)","(4,1)","(4,2)","(4,3)),((5,1)","(5,2)","(5,3)","(6,1)","(6,2)","(6,3)):}}`
`impliesn(F)=18`
`impliesEnnF={(5,3),(6,2)}impliesn(EnnF)=2`
Now `P(E)=5/36`
`P(F)=18/36=1/2` and `P(EnnF)=2/36=1/18`
`:.` Required probability
`=P(E/f)=(P(EnnF))/(P(F))=(1/18)/(1/2)=1/18xx2/1=1/9`