Math 350 Probability – Exam 1 – Fall 2007
Instructions:Answer each question completely and show all work.
1. You are dealt 3 cards from a standard deck. What is the probability that you either have at least
two aces or at least two kings?
2. A fair k-sided die is one in which the numbers {1,2, . . . , k}are rolled with equal probability.
Suppose fair 4,6,8,12 and 20-sided dice are placed in a bag and selected at random (each of the
5 dice is equally likely to be selected) and then that die is rolled.
(a) What is the probability a 7 is rolled?
(b) What is the conditional probability that the 8-sided die was selected given that a 7 is rolled?
(c) Suppose that a die selected from the bag and rolled 3 times (so the same die is rolled 3
times). What is the probability that one of the 4-sided dice was chosen given that all 3 rolls
are between 1 and 4?
3. In a certain community, 36% of the families own a dog, and 22% of the familes that own a dog
also own a cat. In addition, 30% of families own a cat.
What is:
(a) the probability that a randomly selected family owns both a dog and a cat
(b) the probability that a randomly selected family owns a dog given that it owns a cat?
4. Suppose that events E1,E2and E3are independent with P(E1) = p1, P(E2) = p2and P(E3) = p3.
Further suppose that E4is disjoint with each of E1, E2, E3and P(E4) = p4.
(a) Find the probability of that:
i. both E1and E2occur but not E3.
ii. at least one of the events E1, E2or E3occurs.
iii. at least one of the events E1, E2, E3or E4occurs.
iv. exactly two of the four events occur.
(b) Give an example of an experiment and four events E1,E2, E3and E4that satisfy the above
conditions.
5. Prove that if A, B, C, D are independent events then the events A∩Band C∩Dare independent.
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