A die is rolled. Let $E$ be the event "die shows $4$ " and $F$ be the event "die shows even number". Are $E$ and $F$ mutually exclusive?

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Solution

When a die is rolled the sample space is given by,
$S = {1, 2, 3, 4, 5, 6}$
Now according to question $E = {4}$ and $F = {2, 4, 6}$
Clearly $E \cap F = {4} \neq ϕ$
Hence, $E$ and $F$ are not mutually exclusive events.

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A die is rolled . Let us define event E_{1} as the set of possible outcomes where the number on the face of the die is even and event E_{2} as the set of possible outcomes where the number on the face of the die is odd . Event E_{1} and E_{2} are mutually exclusive .

Two dice are thrown. The events A, B, C, D, E and F are described as follows :

A = Getting an even number on the first die.

B = Getting an odd number on the first die.

C = Getting at most 5 as sum of the numbers on the two dice.

D = Getting the sum of the numbers on the dice greater than 5 but less than 10.

E = Getting at least 10 as the sum of the numbers on the dice.

F = Getting an odd number on one of the dice.

(i) Describe the following events : A and B, B or C, B and C, A and E, A or F, A and F

(ii) State true or false :

(a) A and B are mutually exclusive.

(b) A and B are mutually exclusive and exhaustive events.

(d) C and D are mutually exclusive and exhaustive events.

(e) C, D and E are mutually exclusive and exhaustive events.

(f) A' and B' are mutually exclusive events.

(g) A, B, F arc mutually exclusive and exhaustive events.

E = {1, 3, 5}

F = {2, 3}

P (E | F)

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