Question

please show work/reasoning if possible

8: An experiment has three possible elementary outcomes a, b, and c. Thus, the sample space is S = {a, b, c}. Define three events E1 = {a, b}, E2 = {a, c}, and E3 = {b, c}.

Let us also define a probability measure to have values P[E1] =2/3, P[E2] =1/3, and P[E3] =1/3.

Which one of the following derivation is wrong (or which one best answer the nature of this question)?

A) P(E1 ∩ E2) = P(a).

B) P(E1 ∩ E3) = P(b).

C) P(E2 ∩ E3) = P(c).

D) P(E1 ∩ E2 ∩ E3) = 0.

E) P(E1 E2 E3) = P(S) =1.

F) P(E1 E2 E3) = P(E1) + P(E2) + P(E3) - P(E1 ∩ E2) - P(E2 ∩ E3) - P(E1 ∩ E3)

+ P(E1 ∩ E2 ∩ E3) = 2/3 + 1/3+ 1/3 –{ P(a) + P(b) +P(c)}+0 = 2/3 + 1/3+ 1/3 – 1 = 1/3.

G) Conclude from E) and F), we have 1 = 1/3.

H) Nothing wrong in the above derivation (From A to G)! The reason why that we get

“1= 1/3” is simply because it is impossible for the probability measure defined to be P[E1] =2/3, P[E2] =1/3, and P[E3] =1/3.

Answer 1