MATH1715 Tutee Revision Questions
MATH1715 Tutee Revision Questions
MATH1715 Tutee Revision Questions
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eing the “odd-man out” situation, all trials independent, and the probability of a “success”<br />
being P. Thus Y ∼ geometric(P). You require E[Y ].<br />
For “fun”, you might like to determine E[Y ] as a function of n. Would you play this<br />
game if you went out with a football team (n = 11)? 1 Would you play this game if you were<br />
with your <strong>MATH1715</strong> tutees (n ≈ 20)? 2<br />
Question: 10/1/2011<br />
I’m sorry, I know I must be bombarding you with emails. I can’t work out whether this<br />
question is telling me conditional probabilities or not. Very ambiguous.<br />
On a multiple-choice exam with five choices for each question, a student either knows the<br />
answer to a question or chooses an answer at random. If the probability that the student<br />
knows a correct answer is 2/3, what is the probability that an answer that was marked as<br />
correct was not chosen randomly? Thanks for all your help!<br />
Response: First write down what you know.<br />
P{Knows correct answer} = 2/3 so P{Does not knows correct answer} = 1/3.<br />
P{Correct|Knows correct answer} = 1, P{Correct|Does not know correct answer} = 1/5.<br />
We are asked to obtain P{Knows correct answer|Correct}. Clearly we need to use<br />
Bayes’s theorem.<br />
The way the question is put makes it difficult at first glance to determine what is asked.<br />
But notice that the phrase “that was marked as correct” is what has happened, it is known<br />
or given. What the question is asking is what is the probability the answer “was not chosen<br />
randomly”, in other words what is the probability the correct answer was known by the<br />
student, given that it was marked as correct.<br />
Question: 10/1/2011<br />
An elevator in a building starts with n people and stops at N floors, with n < N. If each<br />
passenger is equally likely to get off at any floor, independently of the others, what is the<br />
probability that at least two passengers get off at the same floor?<br />
Response: You have done this already! Recall Homeworks 1, question 1.6. Remember “a<br />
bus starts with 6 people and stops at 10 different stops”? At least two get off at the same<br />
floor is the complementary event to no-one gets off at the same floor.<br />
Question: 10/1/2011<br />
Suppose that a random variable X has an exponential distribution with parameter λ = 2<br />
and let Y = e −X . What is the set of possible values of the random variable Y , and how<br />
do I find the cumulative distribution function FY (y) = P{Y ≤ y}. How would I obtain the<br />
probability density function fY (y) and the mean E[Y ]? If Y1 is the first digit in a decimal<br />
expansion of Y , what is the probability that Y1 = 6?<br />
Response: I strongly doubt any such transformation question will come up!<br />
1 Only if I wanted to waste 15 minutes, assuming each collective toss takes about ten seconds.<br />
2 Only if we could keep awake for three consecutive days!<br />
2