# THEORETICAL STATISTICS

1. Let A be any event with P(A)> 0, and let B1, B2, . . ., Bk be any set of k mutually exclusive events.
Prove that
P(∪ki=1Bi|A) = ∑ki=1 P(Bi|A)
2. Suppose you toss two fair six-sided dice, Die 1 and Die 2. Let A be the event that the sum of the dots
on the two dice is at least five. Let B be the event that the sum is 7. Let C be the event that Die 1
shows a five.
b) Are any two of these events mutually exclusive? Justify your answer.
c) Is any one of these events independent of the intersection of the other two? Justify your answer.
d) Is any one of these events mutually exclusive from the intersection of the other two? Justify your
e) For three rolls of the pair of dice, find the probability that A occurs exactly once.
3. A truth serum given to a suspect is known to be 80% reliable when the person is guilty and 95% reliable
when the person is innocent. In other words, 20% of the guilty are judged innocent by the serum,
and 5% of the innocent are judged guilty. If the suspect was selected from a group of suspects of
which only 10% have ever committed a crime, and the serum indicates he is innocent, what is the
probability that he is guilty?
4. Suppose that when a U.S. resident visits Europe for the first time, the probability they will see London
is 0.70, that they will see Paris is 0.64, and that they will see Rome is 0.58. The probability they see
both London and Paris is 0.45, both London and Rome is 0.42, and both Paris and Rome is 0.35.
The probability that they will see all three of these cities is 0.23. Find the probability that a U.S.
resident who visits Europe for the first time will
a) see Paris given that they see Rome.
b) see Paris given they see London and Rome.
c) will not see Paris given that they will see at least one of London and Rome.
d) will see none of these three cities.
5. Each of four persons fires one shot at a target. Let Ai be the event that person i hits the target. If the
Ai
’s are independent, and P(A1)=P(A2)=0.7, P(A3)=0.5, and P(A4)=0.4, find the probability that
a) they all hit the target.
b) exactly one hits the target.
6. A hand of 5 cards is to be dealt at random and without replacement from an ordinary deck of playing
cards. Find the probability that the hand contains
a) at least four red cards.
b) at least three diamonds, given that it contains at least four red cards.
7. Machines I, II, and III in a factory all produce springs. The proportions of defective springs from the
machines are 1%, 4%, and 2% respectively. The proportions of total spring production from the
three machines are 30%, 25%, and 45% respectively.
a) what percentage of springs produced in this factory is defective?
b) if a randomly selected spring is defective, what is the probability that it was produced by machine II?

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