Download Counting Outcomes - Olean Middle School

Counting Outcomes (Day 2)
The Fundamental Counting Principle
1
Using a table
The table below shows all of the possible outcomes for rolling two number
cubes.
Going across the top are the outcomes for rolling the first number cube.
Going down the left are all the outcomes for rolling the second number
cube.
Shade all of the possible outcomes that are doubles.
The probability of rolling doubles is
6 or 1
36
6
1
2
3
4
5
6
1
(1,1)
(2,1)
(3,1)
(4,1)
(5,1)
(6,1)
2
(1,2)
(2,2)
(3,2)
(4,2)
(5,2)
(6,2)
3
(1,3)
(2,3)
(3,3)
(4,3)
(5,3)
(6,3)
4
(1,4)
(2,4)
(3,4)
(4,4)
(5,4)
(6,4)
5
(1,5)
(2,5)
(3,5)
(4,5)
(5,5)
(6,5)
6
(1,6)
(2,6)
(3,6)
(4,6)
(5,6)
(6,6)
2
Make your own table.
• Make a table showing all of the possibly outcomes
for flipping two integer chips. Each integer chip has
one side that is red, and one side that is yellow.
• What is the probability of flipping the chips and
getting the same color? 2 1
4

2
Red
Yellow
Red
(red, red)
(red, yellow)
Yellow
(yellow, red)
(yellow, yellow)
3
Fundamental Counting Principle
• If event M can occur in m ways and is followed by
event N that can occur in n ways, then the event
M followed by the event N can occur m(n) ways.
– Example: If a number cube is rolled and a coin is
tossed, there are 6(2) outcomes, or 12 possible
outcomes.
• The fundamental counting principle can be used to
find the total number of outcomes possible when 2
or more events occur.
• It does not create a list of the possible outcomes.
4
Example 1
• In the United States, radio and television stations use
call letters that start with K or W. How many different
call letters with four letters are possible? Use the
fundamental counting principle to solve this question.
Number of
Number of
Number of
possible letters X possible letters X possible letters
for the first
for the second
for the third
letter
letter
letter
2
X
26
X
26
Number of
X possible letters
for the fourth
letter
X
26
There are 35,152 possible call
letters for radio and tv stations.
=
Total number
= of possible call
letters
35,152
5
Example 2
• What is the probability of winning a lottery game
where the winning number is made up of three digits
from 0 to 9 chosen at random? Use the fundamental
counting principle to find the number of possible
outcomes, and write the probability as a fraction.
10 × 10 × 10 = 1,000
There are 1,000 possible outcomes.
However, there is only 1 winning number.
The probability of winning with one ticket is
.
6
• Practice 1: At a pizza parlor, you can choose from 3 types
of crust, 2 types of cheese, and 4 toppings. How many
different one-cheese and one-topping pizzas can be
ordered?
Crusts
Cheese
Toppings
3(2)(4)=24
Regular
Deep dish
Thin
Mozzarella
Feta
Pepperoni
Sausage
Mushroom
Peppers
There are 24 different one
cheese, and one topping
pizzas that can be ordered.
• Practice 2: A map shows three towns. There are 4
roads between towns A and B, and there are 5 roads
between towns B and C. How many different routes
can you travel from town A to town B to town C?
4 (5) =20
There are 20 different routes
you can travel.
7
•Practice 3: Find the number of possible choices for a
2-digit number that is greater than 19. The first number
bigger than 19 is 20.
8 (10) =80
The first digit has to be
greater than 2, so there are
8 numbers between 0-9 that
are greater than 2
Since 20 is greater than 19,
the second digit in the
number can be numbers all
of the numbers 0-9.
There are 80 twodigit numbers
that are greater
than 19.
• Practice 4: Find the number of possible choices for a
4-digit PIN number if the digits cannot be repeated.
10 (9)(8) (7)=5,040
The second digit
The first digit has to be one less
number, because
can be any
number 0-9. the pin can’t
repeat.
There are 5,040 possible choices for
a 4-digit PIN number where the
digits don’t repeat.
The third digit has to be
one less than the
second, because the pin
can’t repeat.
The last digit has to be
one less than the third,
because the pin can’t
8
repeat.