Download The Fundamental Counting Principle

Warm Up
1) In your own words define probability.
2) If you toss a coin 10 times, how many times
SHOULD if come up heads?
3) How did you prepare for your test yesterday?
How do you think you did?
Probability:
• The likelihood that an event will happen
• Compare the chance that a specific event will
happen to all the possible events that could
happen
Some Terms!
• Theoretical probability vs. experimental
probability
– What should happen vs. what actually happens
• Outcome: the result of an experiment
• Sample Space: all the possible outcomes of an
experiment
• Trial: one iteration of an experiment
Take out a coin!
Heads
10 Flips
Total
1 Flip per
student
10 Flips
per
student
Tails
Total
Probability
Law of Large Numbers
As more trials are included in an experiment,
the experimental probability of a particular
outcome more closely approaches the
theoretical probability of that outcome.
Fundamental Counting Principle
•
•
•
•
Suppose that two events occur in order
The first can occur in m ways
The second can occur in n ways
Then the two events can occur in m x n ways
# of ways
the first
event can
occur
# of ways
the second
event can
occur
# of ways
the third
event can
occur
Keep doing
this for all
the events
that occur
Example 1
• A fast food restaurant sells hot dogs,
hamburgers, chicken sandwiches, and
barbecue sandwiches. They offer as sides
french fries, hushpuppies, or onion rings.
• How many possible combinations are
there?
__________________
Event 1: Choose an
entrée.
__________________
Event 2: Choose a side.
Multiply those bad
boys.
Example 2
• A mechanic offers three types oil
changes (standard, synthetic, and
high mileage), two types of wiper
blades (low profile, and heavy use)
and two types of mufflers (chrome
and matte black).
• How many possible combinations are
there?
Example 3
An ice cream store offers three
types of cones and 31 flavors.
How many different single-cone
ice-cream cones is it possible to
buy at this store?
Example 4: A little more challenging!
In a certain state, automobile
license plates display three
letters followed by three
digits. How many such
plates are possible if
repetition of the letter is
allowed?
Solution
26
26
26
10
10
By the Fundamental Counting Principle, the
number of possible license plates is…
26 x 26 x 26 x 10 x 10 x 10
17, 576, 000
10
Example 5: Stepping it up!
In a certain state, automobile
license plates display three
letters followed by three digits.
How many such plates are
possible if repetition of the
letters is not allowed?
Solution
26
25
24
10
10
10
By the Fundamental Counting Principle, the
number of possible license plates is…
26 x 25 x 24 x 10 x 10 x 10
15, 600, 000
Factorial Notation!
• Product of n and each previous natural
number (integers greater than 0).
• Denoted by n! and is called “n factorial”
3! = 3 x 2 x 1
8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
Example 6
In how many different
ways can a race with six
runners be completed?
Assume there is no tie.
Solution
•
•
•
•
Six possible choices for first place
Five possible choices for second place
Four choices for third place and so on…
So, by the Fundamental Counting Principle the
number of different ways the race can be
completed is…
6 x 5 x 4 x 3 x 2 x 1 = 6! = 720