7 Suppose that a sample space has five equally likely experi

7. Suppose that a sample space has five equally likely experimental outcomes: E1, E2, E3, E4, E5. Let A = {E1,E2} B= {E3,E4) C= {E2,E3,E5} a. Find P(A), P(B), and P(C). b. Find P(A U B). Are A and B mutually exclusive? c. Find A^c, C^c, P(A^c), and P(C^c). d. Find A U B^c and P(A U B^c). e. Find P(BUC).

Solution

Let the sample space S be S=E₁, E₂, E₃, E₄, E₅

a) Since A has two elements, so P(A)=n(A)/n(S)

=2/5

B has two elements so

P(B)

=n(B)/n(S)

=2/5

C has three elements so

P(C)

=n(C)/n(S)

=3/5

b) AUB={ E₁, E₂, E₃, E₄ }

n(AUB)=4

P(AUB)

=n(AUB)/n(S)

=4/5

A and B have no elements in common so they are mutually exclucive events.

c) A^c = S-A

={E₁, E₂, E₃, E₄, E₅} - {E₁, E₂}

={E₃, E₄, E₅}

P(A^c)

=n(A^c)/n(S)

=3/5

C^c=S-C

={E₁, E₂, E₃, E₄, E₅} - {E_2, E₃, E₅}

={E_1, E₄}

P(C^c)

=n(C^c)/n(S)

=2/5

d) B^c

^=S-B

={E₁, E₂, E₃, E₄, E₅} - {E₃, E₄}

={E₁, E₂, E₅}

AUB^c

={E₁, E₂} U {E₁, E₂, E₅}

={E₁, E₂, E₅}

P(AUB^c)

=n(AUB^c)/n(S)

=3/5

e) BUC={E₂, E₃, E₄, E₅}

P(BUC)

=n(BUC)/n(S)

=4/5