Probability Theory

Probability Theory and Random Process1. A fair die is tossed n times and each time the number appearing on the upward face is recorded.
What is the probability that the sum of numbers appearing on upward faces is n + 2? (The answer,of course, is not a number but rather an expression involving n).2. Alice has n keys, of which one will open her door.
(a) If she tries the keys at random, discarding those that do not work, show that the probability
that she will open the door on her kth attempt (where 1 ≤ k ≤ n) is 1/n.
(b) You should not be surprised that the above probably does not depend on k. If you are, then
maybe you were thinking of a different question, namely “What is the probability that Alice
will find the key in her first k attempts?”. Show that this probability is k/n.
(c) Suppose we change the settings, and Alice does not discard previously tried keys – obviously, a
bad strategy. Can we still answer the questions in parts (a) and (b)? Yes (this will be in a future
homework assignment)! But we can no longer deploy counting methods. The reason is simple:
now the sample space is not finite. Describe the sample space for this new experiment.
3. Sometimes order matters, but sometimes it does not! An urn contains n balls, of which
one is special. Of these n balls, k is to be randomly selected, where 1 < k < n. We consider two
different ways of randomly choosing balls from the urn:
(a) k balls are simultaneously withdrawn, with each ball in the urn being equally likely to be picked;
(b) the k balls are withdrawn one at a time, with each selection being equally likely to be any of
the balls that remain at the time.
For each of the two sampling methods, show that the probability that the special ball is selected is
the same, and equal to k
n.
To answer part (a) you have to define the sample space in a such way that the order in which the
k balls are selected does not matter. In part (b) you instead need to account for the order in which
the balls are withdrawn.
4. A pair of dice is rolled until a sum of 5 or 7 appears. Find the probability that a 5 occurs first. Hint:
Let A denote the event that a 5 occurs first and let An be the event that a 5 occurs on the nth roll
and no 5 or 7 occurs on the first n − 1 rolls. Then A = ∪nAn. To compute each P(An) you may
assume that, for every n > 1, each of the 36n possible outcomes of the experiment of rolling the pair
of dice n times are equally likely.
5. Your statistics teacher announces a thirty-page assignment on Monday that is to be finished by
Friday. You intend to read the first n1 pages on Monday, the next n2 pages on Tuesday, the next n3
pages on Wednesday, and the final n4 pages on Thursday, where n1 + n2 + n3 + n4 = 30 and each
ni ≥ 1. In how many ways can you complete the assignment?
1
6. Suppose that n people are to be randomly arranged. Two of them, Alice and Bob are friends and
hope to be seated next to each other.
(a) What is the probability that Alice and Bob end up sitting next to each other if the n people
are arranged in a line?
(b) What would the probability be if the people are randomly arranged in a circle?
7. A bin of 50 manufactured parts contains three defective parts and 47 non-defective parts. A sample of
six parts is selected from the 50 parts. Write down an expression for the number of different samples
of size six that contain exactly two defective parts.
8. An elevator in a building starts with five passengers and stops at seven floors. If every passenger is
equally likely to get off at each floor and all the passengers leave independently of each other, what
is the probability that no two passengers will get off at the same floor? For more information on Probability Theory check on this: https://en.wikipedia.org/wiki/Probability_theory

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