Determine the probability of the following events

Question

Assume that your sample space is S = { O1, O2, O3, O4, O5 } with associated weights (probabilities) w1 = 0.15, w2 = 0.30, w3 = 0.20, w4 = 0.10, and w5 = 0.25. Determine the probability of the following events:
(1) { O2, O5, O3, O1 }:

(2) S:

(3) { O5 }′:

(4) { O4, O2, O1 }∪{ O3, O4 }:

(5) { O2 } ∪{ O3 }:

(6) { O4 } ∩{ O1 }

Answer 1

Given sample space S = {O1, O2, O3, O4, O5} and Pr[O1] = 0.15, Pr[O2] = 0.30, Pr[O3] = 0.20, Pr[O4] = 0.10, and Pr[O5] = 0.25.

Part (1): The probability of a subset of the sample space is simply the sum of the probabilities of each element in the subset. Therefore:

Pr[ {O2, O5, O3, O1}] = 0.30 + 0.25 + 0.20 + 0.15 = 0.90.

Part (2): For any sample space S, Pr[S] = 1.

Part (3): For any set A we have Pr[A'] = 1 - Pr[A]. Therefore:

Pr[ {O5}' ] = 1 - Pr[{O5}] = 1 - 0.25 = 0.75

Part (4): Since {O4, O2, O1} ∪ {O3, O4} = {O1, O2, O3, O4} = {O5}', we have

Pr[{O4, O2, O1} ∪ {O3, O4}] = Pr[ {O5}' ] = 0.75 (from Part 3).

Part (5): Pr[{O2} ∪ {O3}] = Pr[{O2, O3}] = 0.3 + 0.2 = 0.5

Part (6): Since {O4} ∩ {O1} = { } = ∅, and Pr[∅] = 0, the answer here is 0.