# Counting and Probability

MATH 193 Summary of Counting and Probability Techniques
Counting Techniques
 for questions involving “
or
”:
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
OR use a Venn diagram
 when there are multiple choices to make and the order matters (e.g. passwords),
use the multiplication principle:
(# of ways to make the 1st choice)&middot;(# of ways to make the 2nd choice)&middot; &middot; &middot;
 when there are multiple choices to make, the order doesn’t matter, and the same object can’t
be selected twice, use combinations:
nCr = # of ways to choose r objects from a collection of n objects
 when there are multiple choices to make, the order matters, and the same object can’t be
selected twice, use permutations (or the multiplication principle):
nPr = # of ways to choose and order r objects from a collection of n objects
Probability
 P (E) =
the # of outcomes in E
n(E)
=
n(S)
the total # of possible outcomes of the experiment
to find n(E) and n(S):
– find S (the list of all possible outcomes) then identify which outcomes are in E, and
simply count the # of outcomes in each set
– use an appropriate counting technique
 if n(E) is hard to find, but the opposite of E is easier:
P (E) = 1 − P (E ′ )
Examples Involving Choosing/Ordering Two Numbers from {1, 2, 3}:
1. List all of the possible 2-digit PINs using only the numbers 1, 2, 3 if repetition is allowed.
2. List all of the possible 2-digit PINs using only the numbers 1, 2, 3 if repetition is not allowed.
3. List all of the possible ways to choose 2 distinct numbers from {1, 2, 3} if the order doesn’t
matter (i.e. the selection {1, 2} is the same as {2, 1}).
Example: Suppose that a box contains 6 cameras and that 4 of them are defective. A sample of
2 cameras is selected at random without replacement.
Find the probability distribution for X = the number of defective cameras in the sample.