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1.017/1.010 Examples of Combinatorial Probability
Derivations
Example 1
Suppose we toss a die twice.
Then the number of possible outcomes is the product of
n
1
= 6 outcomes on the first toss and
n
1
= 6 outcomes on the second toss, or
n
1
n
2
= 36.
Now
define the event
A
to correspond to
exactly
one 3 in 2 tosses.
This can occur in two
ways.
First, we can obtain a 3 on the first toss (
n
1
=
1, because there is only one way to
achieve a 3 on the first toss) but not on the second (
n
5
=
5 because there are 5 ways to not
obtain a 3 on the second toss), giving
n
1
n
2
= 5.
Alternatively, we can obtain a 3 on the
second toss but not on the first, giving another
n
1
n
2
= 5 outcomes, for a total of 10
outcomes in
A
.
Compute
P
(
A
) from:
278
.
0
36
10
)
(
)
(
)
(
=
=
=
S
n
A
n
A
P
The same result can be obtained by enumerating all 36 outcomes of two die tosses (e.g.
on a tree) and then counting the number with exactly one 3 (do this to check the above
result).
This die toss experiment is an example of
sampling with replacement
, because
outcomes are drawn from the same set of 6 possibilities on each toss (or subexperiment).
Example 2
Suppose you draw 2 letters at random from a set of 4 (say A, B, C, and D).
Each
successive draw is from a smaller set of possible outcomes (once the A has been drawn it
cannot be drawn again and there is one less possible outcome for the second draw).
If the
order
of the letters is relevant, it follows from the product rule that the number of
possible letter pairs is:
The 12 ordered pairs produced by this experiment are:
{AB, AC, AD, BA, BC, BD, CA, CB, CD, DA, DB, DC}
More generally, an
ordered
subset of
k
objects drawn from a set of
n
objects is called a
permutation
of
n
objects taken
k
at a time.
The number of permutations of
n
objects
taken
k
at a time is written
P
k,n
:
)!
(
!
)
1
)...(
2
)(
1
(
,
k
n
n
k
n
n
n
n
P
n
k
−
=
+
−
−
−
=
1
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View Full Document This number can be used to count simple events that result from sampling without
replacement experiments where sampling order is important.
The MATLAB function
factorial(n)
can be used to evaluate
P
k,n
for relatively small
n
.
If the
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This note was uploaded on 11/29/2011 for the course CIVIL 1.00 taught by Professor Georgekocur during the Spring '05 term at MIT.
 Spring '05
 GeorgeKocur

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