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STAT 225: Introduction to Probability Models Extra Examples
1 Introduction to Probability

Extra 1.1 Let A be an event that happens 40% of the time. Let B be an event that happens 75% of the time. Answer the following 4 questions. What is the smallest probability the intersection of A and B can have? What is the largest probability the intersection of A and B can have? What is the smallest probability the union of A and B can have? What is the largest probability the union of A and B can have? .15, .4, .75, and 1 respectively. Extra 1.2 Suppose we are rolling 2 independent, fair 10-sided die. Let A be the event that the sum of the rolls is a prime number. Let B be the event that the sum of the rolls is odd. Let C be the event that the sum of the rolls is even. Find the following sets: BC , A ∩ B, A ∪ C, A ∩ C, (A ∩ C)C , A ∩ (B ∩ C)C , (A ∩ (B ∩ C))C ? BC = C, A ∩ B = A \ {2}, A ∪ C = Ω \ {9, 15}, A ∩ C = {2}, (A ∩ C)C =Ω \ {2}, A ∩ (B ∩ C)C = A, and (A ∩ (B ∩ C))C = Ω. Extra 1.3 Suppose a lottery has balls numbered 1-20. 4 balls are picked at random and without replacement. Let A be the event that all 4 balls are even. Let B be the event that all 4 balls are less than 10. Let C be the event that all 4 balls are primes. (Allow 1 to be a prime.) Find P(A), P(B), and P(C). (Hint an extended general multiplication rule could be helpful.) P(all 4 even) =
(10)4 (20)4

= .0433.

(9) P(all less than 10) = (20)44 = .0260. What numbers are prime? 1, 2, 3, 5, 7, 11, 13, 17, and 19. That is 9 numbers total. P(all (9) prime) = (20)44 =.0260.

Extra 1.4 A school has 100 students. The school offers only 3 language classes, namely Spanish, Italian and Russian. 50 students do not take a language. The Spanish, Italian, and Russian classes have 28, 26, and 16 students respectively. However, 12 students take both Spanish and Italian, 4 students take both Spanish and Russian,

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