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Suppose that we have a sample space S = {E 1, E 2, E 3, E 4, E 5, E 6, E 7}, where E 1, E 2, ..., E 7 denote the sample points. The following probability assignments apply: P(E 1) = 0.1, P(E 2) = 0.15, P(E 3) = 0.15, P(E 4) = 0.25, P(E 5) = 0.15, P(E 6) = 0.05, and P(E 7) = 0.15. Assume the following events when answering the questions. Find P(A), P(B), and P(C) (to 2 decimals). P(A) ____ P(B) ____ P(C) ____ What is P(A B) (to 2 decimals)? ____ What is P(A B) (to 2 decimals)? ____ Are events A and C mutually exclusive? _________________ What is P(B c ) (to 2 decimals)? ____

Answer :

This question has some data missing so I will first complete the data and than solve this problem

Given:

P(E 1) = 0.1, P(E 2) = 0.15, P(E 3) = 0.15, P(E 4) = 0.25, P(E 5) = 0.15, P(E 6) = 0.05, and P(E 7) = 0.15

and now the missing data which is

A = {E1, E4, E6}

B = {E2, E4, E7}

C = {E2, E3, E5, E7}

Answer:

a) Find P(A), P(B) and P(C)

As we know probability of all the events is the sum of probabilities of all events, so

P(A) = P(E1) + P(E4) +P(E6) = 0.1+0.25+0.05 = 0.40

P(B) = P(E2) + P(E4) +P(E7) = 0.15+0.25+0.15 = 0.75

P(C) = P(E2) + P(E3) +P(E5)+P(E7) = 0.15+0.15+0.15+0.15 = 0.60

b) Find P(AUB),

At first, we will find AUB which is

AUB = {E1, E2, E4, E6, E7}

Now, P(AUB)

P(AUB) = P(E1) + P(E2) +P(E4)+P(E6) +P(E7) = 0.1+0.15+0.25+0.05+0.15

P(AUB) = 0.70

c) Find P(A∩B),

At first, we will find A∩B which is

A∩B = {E4}

Now, P(AUB)

P(A∩B) = P(E4) = 0.25

d) Find A and C are mutually exclusive,

Two events are mutually exclusive if they do not have any outcome in common, and as we can see from A and C, no events are common between them,

So A and C are mutually exclusive

e) Find [tex]B^{c}[/tex] and P([tex]B^{c}[/tex])

[tex]B^{c}[/tex] represents the complement of B, which is defined as all the events which are not in B,

[tex]B^{c}[/tex] =  {E1, E3, E5, E6}, and Now

P([tex]B^{c}[/tex]) = P(E1) + P(E3) +P(E5)+P(E6) = 0.1+0.15+0.15+0.05 = 0.45

P([tex]B^{c}[/tex]) = 0.45

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