UNIT 4: Applications of Probability

UNIT QUESTION: How do you calculate the probability of an event?

Today’s Question:What is a permutation or combination and how do we use it to solve problems?

Let’s work on some definitionsLet’s work on some definitions

Experiment-Experiment- is a situation involving chance that leads to is a situation involving chance that leads to results called outcomes.results called outcomes.

An An outcome outcome is the result of a single trial of an experimentis the result of a single trial of an experiment

An An eventevent is one or more outcomes of an experiment. is one or more outcomes of an experiment.

Probability Probability is the measure of how likely an event is.is the measure of how likely an event is.

Probability of an eventProbability of an event

The probability of event A is the number of The probability of event A is the number of ways event A can occur divided by the total ways event A can occur divided by the total number of possible outcomes.number of possible outcomes.

P(A)=P(A)=The number of ways an event can occurThe number of ways an event can occur

Total number of possible outcomesTotal number of possible outcomes

If P = 0, then the event _______ occur.

ProbabilityProbability

If P = 1, then the event _____ occur.

It is ________

It is ______

So probability is always a number between ____ and ____.

impossible

cannot

certain

must

10

All of the probabilities must add up to 100% or 1.0 in decimal form.

ComplementsComplements

Example: Classroom

P (picking a boy) = 0.60

P (picking a girl) = ____0.40

A glass jar contains A glass jar contains 6 red6 red, , 5 green5 green, , 8 8 blueblue and and 3 yellow3 yellow marbles. marbles.

Experiment: A marble chosen at Experiment: A marble chosen at randomrandom..

Possible outcomesPossible outcomes: choosing a red, blue, green : choosing a red, blue, green or yellow marble.or yellow marble.

Probabilities:Probabilities:

P(red) = P(red) = number of ways to choose red number of ways to choose red = = 6 6 = = 3 3

total number of marbles 22 11total number of marbles 22 11

P(green)= ?, P(P(green)= ?, P(blueblue)= ?, P()= ?, P(yellowyellow)= ?)= ?

There are 3 ways to roll an odd number: 1, 3, 5.

You roll a six-sided die whose sides are numbered from 1 through 6. What is the probability of rolling an ODD number?

Ex. Ex.

P 12

=36

=

Tree Diagrams• Tree diagrams allow

us to see all possible outcomes of an event and calculate their probabilities.

• This tree diagram shows the probabilities of results of flipping three coins.

Use an appropriate method to find the number of outcomes in each of the following situations:

1. Your school cafeteria offers chicken or tuna sandwiches; chips or fruit; and milk, apple juice, or orange juice. If you purchase one sandwich, one side item and one drink, how many different lunches can you choose?

Sandwich(2) Side Item(2) Drink(3) Outcomes

chicken

tuna

There are 12 possible lunches.

chips

fruit

chips

fruit

apple juice orange juice milkapple juice orange juice milk

apple juice orange juice milkapple juice orange juice milk

chicken, chips, apple chicken, chips, orange chicken, chips, milkchicken, fruit, apple chicken, fruit, orange chicken, fruit, milk

tuna, chips, apple tuna, chips, orange tuna, chips, milktuna, fruit, apple tuna, fruit, orange tuna, fruit, milk

Multiplication Counting Principle• At a sporting goods store, skateboards

are available in 8 different deck designs. Each deck design is available with 4 different wheel assemblies. How many skateboard choices does the store offer?

32

Multiplication Counting Principle• A father takes his son Tanner to Wendy’s

for lunch. He tells Tanner he can get the 5 piece nuggets, a spicy chicken sandwich, or a single for the main entrée. For sides: he can get fries, a side salad, potato, or chili. And for drinks: he can get milk, coke, sprite, or the orange drink. How many options for meals does Tanner have?

48

Your iPod players can vary the order in which songs are played. Your iPod currently only contains 8 songs (if you’re a loser). Find the number of orders in which the songs can be played.

1st Song 2nd 3rd 4th 5th 6th 7th 8th Outcomes

There are 40,320 possible song orders.

In this situation it makes more sense to use the Fundamental Counting Principle.

8The solution in this example involves the product of all the integers from n to one (n is representing the starting value). The product of all positive integers less than or equal to a number is a factorial.

• 7 • 6 • 5 • 4 • 3 • 2 • 1 = 40,320

Factorial

EXAMPLE with Songs ‘eight factorial’

The product of counting numbers beginning at n and counting backward to 1 is written n! and it’s called n factorial.factorial.

8! = 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 = 40,320

FactorialSimplify each expression.

a. 4!

b. 6!

c. For the 8th grade field events there are five teams: Red, Orange, Blue, Green, and Yellow. Each team chooses a runner for lanes one through 5. Find the number of ways to arrange the runners.

4 • 3 • 2 • 1 = 24

6 • 5 • 4 • 3 • 2 • 1 = 720

= 5! = 5 • 4 • 3 • 2 • 1 = 120

5. The student council of 15 members must choose a president, a vice president, a secretary, and a treasurer.

President Vice Secretary Treasurer Outcomes

There are 32,760 permutations for choosing the class officers.

In this situation it makes more sense to use the Fundamental Counting Principle.

15 • 14 13• •12 =32,760

Let’s say the student council members’ names were: Hunter, Bethany, Justin, Madison, Kelsey, Mimi, Taylor, Grace, Maighan, Tori, Alex, Paul, Whitney, Randi, and Dalton. If Hunter, Maighan, Whitney, and Alex are elected, would the order in which they are chosen matter?

President Vice President Secretary Treasurer

Although the same individual students are listed in each example above, the listings are not the same. Each listing indicates a different student holding each office. Therefore we must conclude that the order in which they are chosen matters.

Is Hunter Maighan Whitney Alex

the same as…

Whitney Hunter Alex Maighan?

PermutationWhen deciding who goes 1st, 2nd, etc., order is important.

*Note  if  n = r   then   nPr  =  n!

A permutation  is an arrangement or listing of objects in a specific order. 

The order of the arrangement is very important!! 

The notation for a permutation:       nPr = n  is the total number of objects r is the number of objects selected (wanted)

!

( )!

n

n r

Permutation

Notation

PermutationsSimplify each expression.

a. 12P2

b. 10P4

c. At a school science fair, ribbons are given for first, second, third, and fourth place, There are 20 exhibits in the fair. How many different arrangements of four winning exhibits are possible?

12 • 11 = 132

10 • 9 • 8 • 7 = 5,040

= 20P4 = 20 • 19 • 18 • 17 = 116,280

24

1.Bugs Bunny, King Tut, Mickey Mouse and Daffy Duck are going to the movies (they are best friends). How many different ways can they sit in seats A, B, C, and D below?

2. Coach is picking a captain and co-captain from 15 seniors. How many possibilities does he have if they are all equally likely?

A B C D

210

Combinations

• A selection of objects in which order is not important.

• Example – 8 people pair up to do an assignment. How many different pairs are there?

CombinationsAB AC AD AE AF AG AH

BA BC BD BE BF BG BH

CA CB CD CE CF CG CH

DA DB DC DE DF DG DH

EA EB EC ED EF EG EH

FA FB FC FD FE FG FH

GA GB GC GD GE GF GH

HA HB HC HD HE HF HG

• The number of r-combinations of a set with n elements,

• where n is a positive integer and • r is an integer with 0 <= r <= n, • i.e. the number of combinations of

r objects from n unlike objects is

!

! !n r

nC

r n r

Combinations

Example 1

• How many different ways are there

to select two class representatives

from a class of 20 students?

Solution

• The answer is given by the number

of 2-combinations of a set with 20

elements.• The number of such combinations

is

20 2

20!190

2!18!C

Example 2

From a class of 24, the teacher is randomly selecting 3 to help Mr. Griggers with a project. How many combinations are possible?

24 3

24!2024

3! 21!C

Your turn!For your school pictures, you can choose 4 backgrounds from a list of 10. How many combinations of backdrops are possible?

10 4

10!210

4! 6!C

Clarification on Combinations and Permutations

• "My fruit salad is a combination of apples, grapes and bananas"

We don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad.

Clarification on Combinations and Permutations

• "The combination to the safe is 472".

Now we do care about the order. "724" would not work, nor would "247". It has to be exactly 4-7-2.

To sum it up…

• If the order doesn't matter, then it is a Combination.

• If the order does matter, then it is a Permutation.

A Permutation is an ordered Combination.