Math 425: Practice problems for the first midterm exam
Advice: To better prepare for the exam, go over the "selftest
problems and exercises" at the end of each chapter.
Each of these has a complete solution in the back of the book.
 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?
 Two coins fall "heads up" with probabilities
p_{1} and p_{2}, 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"?
 Consider two events A and B such that
P(A)=0.3 and P(B)=0.4. Find the conditional
probability P(AB) under each of the following conditions:
 A and B are independent;
 A and B are mutually exclusive;
 A implies B;
 P(AB)=0.1.

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.