Rob Bayer
Math 55 Worksheet
July 30, 2009
Bayes Rule
1. Suppose you have 3 fair dice and 1 loaded die that comes up 6 with probability
1
3
and each other value with
probability
2
15
. If you pick a die from this set at random, roll it twice, and get a 6 twice, what’s the probability
that you picked the loaded die?
2. A test for a certain disease is 99% speciﬁc (that is, 99% of the time the test will be negative for someone
without the disease) and 98% sensitive (ie, 98% of the time the test will be positive for someone who has the
disease).
(a) If you test positve, what’s the probability that you actually have the disease?
(b) If you test positive twice, how likely is it that you have the disease? Assume the test results are
independently distributed
3. Prove the following generalization of Bayes’ Rule:
If
F
1
,F
2
,...,F
n
are disjoint events of positive probability such that
F
1
∪
F
2
∪···∪
F
n
=
S
and
E
is some event
with positive probability, then
p
(
F
1

E
) =
p
(
E
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 Summer '08
 STRAIN
 Probability, Probability theory, Dice, fair dice, Rob Bayer Bayes

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