Chapter 16 Events and Probability Spaces568
the assigned problems contain errors. If you ask a Teaching Assistant (TA) whether
a problem has an error, then they will answer correctly 80 % of the time. This 80 %
accuracy holds regardless of whether or not a problem has an error. Likewise when
you ask a lecturer, but with only 75 % accuracy.
We formulate this as an experiment of choosing one problem randomly and ask-
ing a particular TA and Lecturer about it. Define the following events:
EWWD“the problem has an error,”
TWWD“the TA says the problem has an error,”
LWWD“the lecturer says the problem has an error.”(a)Translate the description above into a precise set of equations involving con-
ditional probabilities among the eventsE,T, andL.
(b)Suppose you have doubts about a problem and ask a TA about it, and they tell
you that the problem is correct. To double-check, you ask a lecturer, who says that
the problem has an error. Assuming thatthe correctness of the lecturers’ answer
and the TA’s answer are independent of each other, regardless of whether there is
an error^8 , what is the probability that there is an error in the problem?
(c)Is the event that “the TA says that there is an error”, independent of the event
that “the lecturer says that there is an error”?
Problem 16.19. (a)Suppose you repeatedly flip a fair coin until you see the se-
quenceHHTor the sequenceTTH. What is the probability you will seeHHTfirst?
Hint:Symmetry between Heads and Tails.
(b)What is the probability you see the sequenceHTTbefore you see the sequence
HHT?Hint:Try to find the probability thatHHTcomes beforeHTTconditioning on
whether you first toss anHor aT. The answer is not1=2.
Problem 16.20.
A 52-card deck is thoroughly shuffled and you are dealt a hand of 13 cards.
(a)If you have one ace, what is the probability that you have a second ace?(b)If you have the ace of spades, what is the probability that you have a second
ace?
(^8) This assumption is questionable: by and large, we would expect the lecturer and the TA’s to spot
the same glaring errors and to be fooled by the same subtle ones.