Question

One die is rolled. Let

A = event the die comes up even
B = event the die comes up odd
C = event the die comes up 4 or more
D = event the die comes up at most 2
E = event the die comes up 3

give your answer as "yes" or "no" (without the quotation)

(a)  Are events A and C mutually exclusive?
answer:

(b)  Are events C, D and E mutually exclusive?
answer:

(c)  Are events C and E mutually exclusive?
answer:

(d)  Are events A and B mutually exclusive?
answer:

(e)  Are there any four mutually exclusive events among A, B, C, D and E?
answer:

Answer 1

Solution:

Given: One die is rolled. Let

A = event the die comes up even = { 2, 4, 6}

B = event the die comes up odd = {1 , 3 , 5 }

C = event the die comes up 4 or more = { 4 , 5 , 6 }

D = event the die comes up at most 2 = { 1 , 2 }

E = event the die comes up 3 = { 3 }

Part (a) Are events A and C mutually exclusive?

A = { 2, 4, 6}

C = { 4 , 5 , 6 }

Since events A and C has two common outcomes 4.6

that is: ( A and B) = { 4 , 6 } , thus events A and C are not mutually exclusive

Thus answer is : No

Part (b)  Are events C, D and E mutually exclusive?

C = { 4 , 5 , 6 }

D = { 1 , 2 }

E = { 3 }

None of the events have any common elements

thus events C, D and E are mutually exclusive

Thus answer is : Yes

Part (c)  Are events C and E mutually exclusive?

C = { 4 , 5 , 6 }

E = { 3 }

Since C and E has no common element, C and E are mutually exclusive

thus answer is: Yes

Part (d)  Are events A and B mutually exclusive?

A = { 2, 4, 6}

B = {1 , 3 , 5 }

Since A and B has no common element, A and B are mutually exclusive

thus answer is: Yes

Part (e)  Are there any four mutually exclusive events among A, B, C, D and E?

i) From part d) we have: A and B are mutually exclusive  

ii) From part c) we have : C and E are mutually exclusive

iii)

C = { 4 , 5 , 6 }

D = { 1 , 2 }

Since C and D has no common element, C and D are mutually exclusive

iv)

D = { 1 , 2 }

E = { 3 }

Since D and E has no common element, D and E are mutually exclusive

Thus we have four mutually exclusive events among A, B, C, D and E

Thus answer is: Yes