Question

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) (ii) P ((E ∪ F)|G) and P ((E ∩ G)|G)

Answer 1

It is given that a die is rolled and the events are given as,

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

When a die is rolled the possible outcomes is 6.

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

The probability of events E, F and G is,

P( E )= 3 6 P( F )= 2 6 P( G )= 4 6

(i)

The common outcomes between E and F are,

( E∩F )={ 3 } P( E∩F )= 1 6

The probability P( E|F ) is calculated as,

P( E|F )= P( E∩F ) P( F ) = 1 6 2 6 = 1 2

The probability P( F|E ) is calculated as,

P( F|E )= P( E∩F ) P( E ) = 1 6 3 6 = 1 3

Therefore, the probability P( E|F ) is 1 2 and P( F|E ) is 1 3 .

(ii)

The common outcomes between E and G are,

( E∩G )={ 3,5 } P( E∩G )= 2 6

The probability P( E|G ) is calculated as,

P( E|G )= P( E∩G ) P( G ) = 2 6 4 6 = 2 4 = 1 2

The probability P( G|E ) is calculated as,

P( G|E )= P( E∩G ) P( E ) = 2 6 3 6 = 2 3

Therefore, the probability P( E|G ) is 1 2 and P( G|E ) is 2 3 .

(iii)

The union of E and F is,

E∪F={ 1,2,3,5 } P( E∪F )= 4 6

The common outcomes between E and F are,

E∩F={ 3 } P( E∩F )= 1 6

The common outcomes between E∪F and G are,

( E∪F )∩G={ 1,2,3,5 }∩{ 2,3,4,5 } ={ 2,3,5 } P( ( E∪F )∩G )= 3 6

The probability P( ( E∪F )|G ) is calculated as,

P( ( E∪F )|G )= P( ( E∪F )∩G ) P( G ) = 3 6 4 6 = 3 4

The common outcomes between E∩F and G are,

( E∩F )∩G={ 3 }∩{ 2,3,4,5 } ={ 3 } P( ( E∩F )∩G )= 1 6

The probability P( ( E∩F )|G ) is calculated as,

P( ( E∩F )|G )= P( ( E∩F )∩G ) P( G ) = 1 6 4 6 = 1 4

Therefore, the probability P( ( E∪F )|G ) is 3 4 and P( ( E∩F )|G ) is 1 4 .