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1. (12 points)
Consider a game where two sixsided dice are being rolled.
(a) What is the total number of atomic events in this game?
Answer:
36.
(b) What is the probability of getting a 4 on one of the two dice?
Answer:
11/36 (not 12/36). There are 11 atomic events with a 4 on one of the two
dice, amongst a total of 36. By the principle of indiﬀerence, it is reasonable to assume
that each atomic event is equally likely, and hence the ﬁnal probability is 11/36. (If
you thought the answer was 12/36, you are double counting the event with 4s on
both dice.)
(c) Now, suppose I tell you that the sum of the two dice is greater than 7. What is the
probability of getting a 4 on one of the two dice now?
Answer:
One can calculate this directly using the deﬁnition of conditional proba
bility. The number of atomic events where the sum of the two dice is greater than 7
is 15 (by explicit counting). The number of events within this set of 15 events with
a 4 is 5 (again by explicit counting). So the joint probability of getting a 4 on one of
the dice as well as sum over 7 is 5/36, and the probability of the sum of the two dice
being greater than 7 is 15/36; the conditional probability of getting a 4 on one of the
dice given that the sum is greater than 7 is 5/15 = 1/3.
One can also calculate this using Bayes’ Rule. We already saw in (b) that the prior
probability of obtaining a 4 on one of the dice is 11/36; now we want the posterior
probability given that the sum is greater than 7. To apply Bayes’ Rule we will need
the data (or evidence) likelihood, which is the probability of sum greater than 7, given
that one of the dice is 4. There are 5 events with sum greater than 7 among the 11
events which have at least one 4 on them; hence the evidence likelihood is 5/11. The
prior probability of the evidence is the probability of sum being greater than 7, which
is 15/36. By Bayes’ rule, posterior probability is prior probability times evidence
likelihood divided by prior probability of evidence. Hence, posterior probability of 4
on one of the two dice given sum is greater than 7 is
11
/
36
*
5
/
11
15
/
36
= 5
/
15 = 1
/
3.
2. (16 points)
After your yearly checkup, the doctor has bad news and good news. The bad news is
that you tested positive for a serious disease and that the test is 99% accurate (i.e., the
probability of testing positive when you do have the disease is 0.99, as is the probability
of testing negative when you don’t have the disease). The good news is that this is a
rare disease, striking only 1 in 10,000 people of your age. What are the chances that you
actually have the disease?
Answer:
First thing we should do is to deﬁne our random variables. Let
T
be the
boolean random variable which is true if the test is positive and false otherwise. Let
D
be the boolean random variable which is true if you have the disease and false otherwise.
1
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This note was uploaded on 03/16/2011 for the course CSE 630 taught by Professor Naeemshareef during the Spring '10 term at Ohio State.
 Spring '10
 NaeemShareef

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