Download If an event M can occur m ways and is followed by an event N that

Fundamental Counting Principle: If an event M can
occur m ways and is followed by an event N that can
occur n ways, then event M followed by event N can
occur m n ways.
Permutation: An arrangement of items in a particular
order. The number of permutations of n items of a set
arranged r items at a time is
n Pr 
n!
(n  r )!
n! n  (n  1)  ...  3  2  1
Combination: A selection in which order does not
matter.
Example: A club has nine members. In how many ways
can a president, vice president, and secretary be chosen
from its members of this club?
Example: From 20 raffle tickets in a hat, four tickets are
to be selected in order. The holder of the first ticket wins
a car, the second a motorcycle, the third a bicycle and the
fourth a skateboard. In how many different ways can
these prizes be awarded?
COMBINATIONS: ORDER DOES NOT MATTER!
The number of combinations of n objects taken r at a time
n!
C 
is: n r r!(n  r )!
Example: 5
C3
Example: A club has nine members. In how many ways
can a committee of three be chosen from the members of
this club?
Example: From 20 raffle tickets in a hat, four tickets are
to be chosen at random. The holders of the winning
tickets are to be awarded free trips to the Bahamas. In
how many ways can the four winners be chosen?
A pizza parlor offers the basic cheese pizza and a choice
of 16 toppings. How many different kinds of pizza can be
ordered at this pizza parlor?
Guidelines for using Permutations and Combinations:
- When we want to find the number of ways of picking
r objects from n objects we need to ask ourselves:
Does the order of the objects matter?
- If the order matters, we use permutations
- If the order doesn’t matter, we use combinations.
Example: A chemistry teacher divides his class into eight
groups. Each group submits one drawing of the molecular
structure of water. He will select four of the drawings to
display. In how many different ways can he select the
drawings?
Example: You will draw winners from a total of 25 tickets
in a raffle. The first ticket wins $100. The second ticket
wins $50. The third ticket wins $10. In how many
different ways can you draw the three winning tickets?
Example: How many 5 card hands can be dealt from a
deck of 52 cards?
Example: A committee of seven consisting of a chairman,
a vice chairman, a secretary, and four other members, is
to be chosen from a class of 20 students. In how many
ways can this committee be chosen?
Probability:
Experimental Probability of an event:
Example: Of the 60 vehicles in a teacher’s parking lot
today, 15 are pickup trucks. What is the experimental
probability that a vehicle in the lot is a pickup truck?
Example: A class tossed coins and recorded 161 heads
and 179 tails. What is the experimental probability of
heads? Of tails?
Theoretical Probability:
- Sample Space: the set of all possible outcomes to an
experiment or activity.
- Equally Likely: When each outcome in a sample
space has the same chance of occurring.
- If a sample space has n equally likely outcomes
and an event A occurs in m of these outcomes,
then the theoretical probability of event A is:
Example: What is the theoretical probability of each
event?
a. Getting a 5 on one roll of a standard number cube
b. Getting a sum of 5 on one roll of two standard
number cubes
Using Combinations to find probability:
Example: A five-card poker hand is drawn from a
standard deck of 52 cards. What is the probability that all
five cards are spades?
Example: A bag contains 20 Frisbees, of which four are
defective. If two Frisbees are selected at random from the
bag, what is the probability that both are defective?
Complement of an Event: The complement of an event
E is the set of outcomes in the sample space that is not in
E. We denote the complement of an event E by E’.
Probability of the complement of an event:
Example: An urn contains 10 red marbles and 15 blue
marbles. Six marbles are drawn at random from the urn.
What is the probability that at least one marble is red?