Transcript The Counting Principle (Multiplication Principle)

The Counting Principle
(Multiplication Principle)
Multiplication principle: the
total number of outcomes for
an event is found by multiplying
the number of choices for each
stage of the event
The multiplication principle
gives you the number of
outcomes NOT the probability.
Examples:
1. At an ice cream shop there are 31 flavors and 25
toppings. How many different ways are there to
make a one-scoop ice cream sundae with one
topping?
31●25 = 775
2. There are 4 quarterbacks and 6 centers on a
football team that has 60 players. How many
quarterback-center pairings are possible?
4●6 = 24
3. You roll three dice, how many outcomes are there?
6●6●6 = 216
4. Three coins are tossed, how many outcomes are
there?
2●2●2=8
5. Canned bean are packed in three sizes:
small, medium and large; and are red,
black-eyed, green, yellow, or baked. How
many size-type outcomes are there?
3●5=15
6. A confectioner offers milk, dark, or white
chocolates with solid, cream, jelly, nut, fruit,
or caramel centers. How many flavor
choices are there?
3●6=18
Tree Diagrams and Counting
• Jennifer and her family went to Dilly’s Deli for
lunch. Jennifer wanted a sandwich. She had
three choices of bread: white, wheat or rye.
She had three choices of meat: turkey, ham or
roast beef. She can choose one type of meat
and one type of bread for her sandwich.
Complete the tree diagram and find the
number of choices that Jennifer has for her
sandwich.
Total choices = 3 breads ● 3 meats = 9 sandwiches
A Tree Diagram can be used to show you the
number of outcomes in an event.
9
choices
Examples:
Cheryl has a choice of a pink, red or yellow blouse
with white or black slacks for an outfit. How many
possible outfits are there?
6
outfits
A coin is tossed and a spinner is spun.
How many outcomes are there?
2
3
1
6
outcomes
Permutations
• With your group find as many
arrangements of the letters A, H, M, T as
you can.
• How many 2 letter arrangements are there?
• Could you do this an easier way?
• Use a tree diagram, or…
Examples: How many different ways can
the letters of each word be arranged?
1. SAND
2. GREEN
4! = 4●3●2●1 = 24
5! = 5●4●3●2●1 = 120
3. CAT
3! = 3●2●1 = 6
Examples: Find the value.
4. 7!
5. P(8,2)
7●6●5●4●3●2●1 = 5040
8●7 = 56
6. P(9, 3)
9●8●7 = 504
Examples:
7. In how many ways can six people line up
for a photograph?
720 ways
6! = 6●5●4●3●2●1
8. A building inspector is supposed to inspect
10 building for safety code violations. In how
many different orders can the inspector visit
the buildings?
10 ! =
10●9●8●7●6●5●4●3●2●1 =
3,628,800 ways
Examples:
9. How many 3 letter words can you make
from 5 letters?
P(5, 3) 5●4●3
60 words
10. How many 4-letter, two digit license plate
numbers can you make?
a. If repeat letters and
numbers allowed
b. If repeat letters and
numbers not allowed
26●26●26●26●10●10
45,697,600 plates
26●25●24●23●10●9
32,292,000 plates
Probability of AND events:
Notation:
P(A and B) = P(A)•P(B)
Probability of AND events, you MULTIPLY!
If you draw a card from a deck numbers 1 through 10 and toss a die, find the
probability of each outcome.
Event A: pick a card
Outcomes when you pick a card:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Event B: roll a die
Outcomes when you roll a die:
1, 2, 3, 4, 5, 6
If you draw a card from a deck numbers 1 through 10 and toss a die, find the
probability of each outcome.
Event A: pick a card
Outcomes: 10
Outcomes: 6
2. two even numbers
1. P(a 10 and a 3)
P(10) = _1_
10
P(3) = _1_
6
P(10 and 3)=
_1_ • _1_ =
10 6
Event B: roll a die
P(even) = _5_ = _1_
10
2
P(even) = _1_
2
P(2even)=
_1_
60
_1_ • _1_ =
2
2
_1_
4
If you draw a card from a deck numbers 1 through 10 and toss a die, find the
probability of each outcome.
Event A: pick a card
Outcomes: 10
3. P(2primenumbers)
P(prime) = _2_
P(prime) = _3_ = _1_
6
2
5
Event B: roll a die
Outcomes: 6
4. two odd numbers
P(odd) = _5_
10
P(odd) = _1_
2
P(prime and prime)=
_2_ • _1_ =
5
2
_1_
P(2odd)=
5
_1_ • _1_ =
2
2
_1_
4
If you draw a card from a deck numbers 1 through 10 and toss a die, find the
probability of each outcome.
Event A: pick a card
Outcomes: 10
5. P(even and prime)
P(even) = _1_
2
P(prime) = _3_ = _1_
6
2
P(prime and prime)=
_1_ • _1_ =
2
2
_1_
4
Event B: roll a die
Outcomes: 6