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Probability Axioms and Counting

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Question1: Probability Axioms
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.

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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

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) ...

Solution provided by:
  • MSc, California State Polytechnic University, Pomona
  • MBA, University of California, Riverside
  • BSc, California State Polytechnic University, Pomona
  • BSc, California State Polytechnic University, Pomona
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