Question1: ProbabilityAxioms

Assume that P(A) = 0.5, P (A ∩ C) = 0.18, P(C) = 0.4, P(B) = 0.4, P (A ∩ B ∩ C) = 0.06,

P (B ∩ C) = 0.18, and P (A ∩ B) = 0.15. Calculate the following probabilities:

a. P (A ∪ B ∪ C)

b. P (A’ ∩ (B ∪ C))

c. P ((B ∩ C)’ ∪ (A ∩ B)’)

d. P (A/ (A ∩ C))

Question 2: Counting

A hand of five cards is chosen randomly and without replacement from a standard deck of 52 cards.

a. What is the probability that the hand contains exactly 2 aces and exactly 1 kings? Include at least 4 digits following the decimal point in your answer.

a)

By inclusion-exclusion principle,

P (A ∪ B ∪ C) = P(A) + P(B) + P(C) – P (A ∩ C) – P (B ∩ C) – P (A ∩ B) + P (A ∩ B ∩ C)

= 0.5 + 0.4 + 0.4 – 0.18 – 0.18 + 0.15 + 0.06

= 0.85

b)

By inclusion-exclusion principle,

P (B ∪ C) = P(B) + P(C) – P (B ∩ C)

= 0.4 + 0.4 – 0.18

= 0.62

P (A’ ∩ (B ∪ C)) = P (B ∪ C) …

Step-by-step solutions are provided in both text form and pdf form. The axioms, laws, principles, formulas or theorems used to arrive at the solution are stated.