Math 411 - Fall 2014 - 1st Exam.
1. (25 pts) Determine which of the following statements are true or false:
– a) If A, B, C are three events suich that
P (ABC) = P (A)P (B)P (C)
then these events are independent.
– b) Let An be a decreasing sequence of events. Then P (∪∞
n=1 An ) = limn→∞ P (An ).
– c) If A, B are two events, then P (AB) ≤ P (A) + P (B).
– d) The number of all subsets of a set with n points is n2 .
– e) Let F1 , F2 , F3 three mutually disjoint events with the property S = F1 ∪ F2 ∪
F3 . Then for eny event E,
P (E) = P (E|F1 )P (F1 ) + P (E|F2 )P (F2 ) + P (E|F3 )P (F3 ).
2. (20 pts)
1. A committee of 4 people is chosen randomly from a group of 12 men and
4 women. Find the probability that at least 2 women and 1 man is in the
committee.
2. A random arrangement of the letters A,B, C, C is given. What is the probability that C will be the first one?
3. (25 pts)
1. Let A, B, C be three independent events with P (A) = 41 , P (B) = 54 and P (C) =
1
. What is the probability that at least one of them will happen?
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2. 4 independent trials, each resulting in a success with probability 0.2, are performed. What is the probability that you will have at least one success.
4. (25 pts) Suppose that we have two identical boxes. We know that one box contains
10 balls numbered 1 to 10 and the other contains 100 balls numbered 1 to 100. We
randomly choose a box and remove one ball. Given that the ball that we took out
is “no 7” what is the probability that the box we choose is the one with 10 balls?
5. (25 pts) Suppose that you randomly put 4 (different) letters into 4 envelopes. Find
the probability that no letter is matched to the correct envelop.
Write 100 out of the 120 available points
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