Download Handout 5 Supplements material in Section 2.4 of L-G

Handout 5
Supplements material in Section 2.4 of L-G
Conditional probability
Example 5.1 Suppose we have two dice, a black one and a white one. Assume we throw the white
die before the black die. What is the probability that the faces sum to 3 (let’s call that event )?
Solution. Before the dice are thrown the probability is . Now suppose the white die comes up 1
(let’s call that event ). What is he probability of now?
Definition 5.1 We will call it the conditional probability that event
tion that even has already occurred. We will write it as:
and say “The probability of
given
will occur, given the condi-
”.
Before any dice were thrown, the sample space had 36 outcomes, but now that the event
occurred, the outcome must belong to the reduced sample space :
has
In this reduced sample space of six elementary outcomes, only one outcome
sums to 3.
Thus,
. In general , to find the conditional probability
we look at the event
and as part of the reduced sample space of . We translate this, now, into a formal definition:
The conditional probability of
given
is
from which you can directly verify some intuitive facts:
(once
occurs it is certain), and also, when
(once
has occurred,
is impossible).
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and
are mutually exclusive,
36-217: Probability Theory and Random Processes
Fall 1997
Rearranging the above definition gives us the multiplication rule:
Example 5.2 Suppose the human population is 50% male and 50% female and further suppose
that 10% of males are color blind and 1% of females are color blind. What is the probability that a
randomly selected person is color blind?
Solution. Let’s write out what we know. Let
that a person is color blind. We know that
be males and
females and also
be the event
and are also given that
What we are looking for is
. Some plain thinking suggests that since 10% of males are
color blind, and males are 50% of the whole population, and since 1% of the rest 50% of the
population are also color blind then
A fancy way to write that is
This expression is a simple version of something called the law of total probability.
Example 5.3 Imagine that the chances that a randomly selected individual is HIV positive are
2/100. Now suppose that if the person is HIV positive, the chances of being tested positive are
95/100. Also assume that the chances of being tested positive if you are HIV negative are 10/100.
In other words
Are +ve
and
Test +ve Are +ve
Test +ve Are -ve
Hence, we also know that
Test -ve Are +ve
Test -ve Are -ve
Taking the definition of conditional probability we get that
Test +ve
Test +ve
Are +ve
Are +ve
Test +ve Are +ve
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Are +ve
36-217: Probability Theory and Random Processes
Fall 1997
How to get answers to sensitive questions without even asking them
Cheating among undergraduates
Suppose I were to ask each one of you if you have ever cheated (in an exam or a homework)
while you were at CMU. I would, probably, NOT get honest answers. Let’s do it, though, the
following way:
Each of you should remember the phone number you had during your last moth of high school.
Let’s call that your home phone number and you can be pretty sure that I don’t know this number
and probably nobody else in the room does. Also think of your social security number.
Now I am going to ask you to give a yes/no answer to what seems like a complicated question.
If your home phone number is even then answer Yes/No to the question: “Have you ever
cheated at CMU?”
If your home number is odd, then answer Yes/No to the question “Is your social security
number even?”
Thus, if your answer was Yes, I don’t know whether you’re telling me that you cheated or you’re
telling me that your social security number is even, since I don’t know your home phone number.
If the proportions of Yes answers is then the estimate of the proportion of people who cheated
in the class is
Proof. Let be the event that a randomly selected person has cheated, and let
be the event that
the answer to the complicated question is Yes. Then from the law of total probability
even phone
even phone
even phone
even phone
odd phone
odd phone
SS even odd phone
odd phone
(1)
We now assume that roughly half of all phone numbers are even and also half of all social security numbers are all even, i.e.,
even phone
odd phone
and
SS even
SS odd
. In addition, let’s assume that whether you cheat or not doesn’t depend on
your phone number, i.e.,
even phone
. Then equation (1) becomes:
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36-217: Probability Theory and Random Processes
Fall 1997
Independence
In the previous “Cheating at CMU” example we assumed that the event cheat at CMU or not
does not depend on your phone number and we wrote that as:
even phone
The aforementioned property is called independence:
Definition 5.2 Two events and are called independent of each other if the occurrence of one
does not affect the occurrence of the other. In other words
(2)
Corollary 5.1 Two events are said to be independent if
Proof.
Example 5.4 Consider again a deck of 52 cards. We are going to pick to cards out of it and we
are going to do it with and without replacement. In both cases we are looking for
1. the probability of drawing two diamonds,
2. the probability of drawing a diamond and a club
Next time: More on independence
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