## Book: Mathematics Part-II

### Chapter: 13. Probability

#### Subject: Maths - Class 12th

##### Q. No. 11 of Exercise 13.1

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11
##### A fair die is rolled. Consider events E = {1,3,5}, F = {2,3} and G = {2,3,4,5}Find(i) P(E|F) and P(F|E)(ii) P(E|G) and P(G|E)(iii) P((E ∪ F)|G) and P ((E ∩ F)|G)

The sample space for the given experiment will be:

S = {1, 2, 3, 4, 5, 6}

Here, E = {1, 3, 5}, F = {2, 3} and G = {2, 3, 4, 5} ……….(I)

……….(II)

Now, E F = {3}, F G = {2, 3}, E G = {3, 5} ……….(III)

……….(IV)

(i) We know that

By definition of conditional probability,

[Using (II) and (IV)]

Similarly, we have

[Using (II) and (IV)]

(ii) We know that

By definition of conditional probability,

Similarly, we have

(iii) Clearly, from (I), we have

E = {1, 3, 5}, F = {2, 3} and G = {2, 3, 4, 5}

E F = {1, 2, 3, 5}

(E F) G = {2, 3, 5}

……….(V)

Now, we know that

By definition of conditional probability,

[Using (II) and (V)]

Similarly, we have E F = {3} [Using (III)]

And G = {2, 3, 4, 5} [Using (I)]

(E F) G = {3}

……….(VI)

So,

[Using (II) and (VI)]

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