MATH 5030, Homework 1
A problem will be due on Friday, August 31, by the end of lecture. The problem will be
announced Thursday afternoon at the course webpage.
Problem 1
a) Let A0 , A1 , . . . An be events of non-zero probability. Show that
P{A0 A1 A2 . . . An } = P{A0 }P{A1 |A0 } · · · P{An |An−1 . . . A0 }
Where are you using the assumption that the events have non-zero probability?
b) You are standing in front of the door that will lead you to Narnia (and you really want
to go there). You have N keys at your disposal and only one of them opens the door.
Every key you try and doesn’t open the door you discard. What is the probability that you
manage to open the door exactly at your k − th try?
Problem 2
Find events A1 , A2 , A3 on some probability space (Ω, P) so that the events are pairwise
independent but not independent.
Problem 3
In answering a question on a multiple choice test, a student either knows the answer (with
probability p) or guesses uniformly at random with probability 1/N where N is the number
of possible answers in every question. Finally, assume the test has 10 questions each worth
1 point. All questions require an answer.
(1) What is the probability that a Christel’s score was 8/10 given that she knew exactly
half the material?
(2) Marton was the only one who answered question 7 correctly. What is the probability
that Marton in fact knew the answer to question 7?
(3) Assume that N = 4. Diane knows a lot of statistics and knows that in multiple
choice tests with four questions, the answer C is correct 40% of the time while
answers A, B, D are correct with equal probability. When Diane guesses, she guesses
according to this scheme. Diane is the only one who answered question 10 correctly.
What is the probability that Diane knew the answer if the correct answer to 10 is
C? What if the answer to question 10 was D?
(4) Assume N = 6. Tommy forgot he had a test that day. He answers 7 of the questions
at random. What is his expected grade? What is the variance of his expected grade?
Assume that wrong answers are penalized by negative score −1/2 point. what is
the expectation and variance of his grade now?
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