MATH 141-501
Section 7.4 Lecture Notes
Use of Counting Techniques in Probability
Question: Suppose we flip a coin 100 times and record the result of each
flip. What is the probability of getting exactly 60 heads?
What are some elements of the sample space?
Example element: THTHHTTTTHTHHTTTTTTHTHTHHTHTTTTHHTTTTTTHHHHTHTHHTTTTHTTTTHHTTHHHTTTTHTH HHTHHTHTHTTHTTTTTTTHHTTTHHHHT
Verbal description: Strings (”Words”) of length 100 made up only of ’H’s
and ’T’s.
How many elements are in the sample space?
Is this sample space uniform?
What is the probability of the event that when we perform this experiment,
we see exactly 60 heads?
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Don’t Hurt Yourself !!
Use Counting Techniques
Recall that if a sample space S uniform ( the probability of each outcome is the same ),
then the probability of an event E is given by
P (E) =
n(E)
.
n(S)
In a uniform sample space, it is often much easier to calculate the number
of elements in an event than to list all the outcomes in that event.
Example - Texas Hold ’Em: Two cards are selected from a well-shuffled
standard deck. What is the probability that the player gets...
... a pair of aces (two aces)?
... any pair (two cards of the same face value)?
...two cards of the same suit?
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Example:
A box contains 200 microchips, of which 40 are substandard. If 3 of the
chips are taken from the box, what is the probability that...
....they are all substandard?
...exactly 2 are substandard?
...at least 1 is substandard?
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Example:
An exam a certain course consists of 10 true-or-false questions. A student
comes in (without studying) and guesses on every question. What is the probability that the student makes...
...an A?
...an F?
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Example:
A ticket for the Powerball Lottery consists of five numbers from 1 to 59 (with
no repeats allowed) represented by white balls, and a number from to 1 to 35
(called the Powerball) represented by a red ball. The Powerball number can be
the same as one of the numbers from the white balls.
How many total possible combinations are there for Powerball? (i.e. what
is the size of the sample space?)
Is the sample space uniform?
For a single ticket, what is the probability of winning Powerball?
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The Birthday Paradox
Question: If ten people are selected at random, what is the probability
that two of them have the same birthday?
Question: How many people need to be selected at random in order for the
probability of two of them to have the same birthday to be greater than 0.50?
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