Math 425: Practice problems for the first midterm exam

Advice: To better prepare for the exam, go over the "self-test problems and exercises" at the end of each chapter. Each of these has a complete solution in the back of the book.

1. Suppose that in a deck of 20 cards, each card has one of the numbers 1, 2, 3, 4, 5, with four cards of each number. Ten cards are chosen from the deck at random.
   • If the cards are drawn with replacement, what is the probability that each of the numbers 1, 2, 3, 4, 5 will appear exactly twice?
   • If the cards are drawn without replacement, what is the probability that each of the numbers 1, 2, 3, 4, 5 will appear exactly twice?
   • If the cards are drawn without replacement, what is the probability that the numbers X, Y, Z on the first three cards satisfy the strict inequalities X<Y<Z?
2. Two coins fall "heads up" with probabilities p₁ and p₂, respectively. Both coins are tossed.
   • What is the probability that they show the same face?
   • If they do show the same face, what is the probability that the face they both show is "heads"?
3. Consider two events A and B such that P(A)=0.3 and P(B)=0.4. Find the conditional probability P(A|B) under each of the following conditions:
   • A and B are independent;
   • A and B are mutually exclusive;
   • A implies B;
   • P(AB)=0.1.
4. A parallel system functions whenever at least one of its components works. Consider a parallel system of 3 components and suppose that these components independently work with probabilities 0.7, 0.8, and 0.9.
   • What is the probability that the system is functioning?
   • Find the conditional probability that component 1 works given that the system is functioning.