Unit 8

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For lunch, in how many different ways could

you choose a sandwich, side dish, and

dessert?

If an event M that can occur in m ways is

followed by event N that can occur in n

ways, then event M followed by event N can

occur in 𝑚 ∙ 𝑛 ways.

Example: 3 pants and 2 shirts give 3 ∙ 2 = 6

possible outfits.

In 1966, one type of Maryland license plate

had two letters followed by four digits. How

many of this type of license plate were

possible?

How many license plates would be possible

if the state did not allow repeated numbers

or letters?

A permutation is an arrangement of items

in a particular order.

Suppose you wanted to find the number of

ways to order three items. There are 3

ways to choose the first, 2 ways to choose

the second, and only 1 way to choose the

last.

3 ∙ 2 ∙ 1 = 6 permutations

Using factorial notation, you can write

3 ∙ 2 ∙ 1 as 3!, read “3 factorial.”

For any positive integer n, n factorial is

𝑛! = 𝑛 𝑛 − 1 ∙ . . .∙ 3 ∙ 2 ∙ 1. Zero factorial is

0! = 1.

How many ways can you file 12 folders, one

after another, in a drawer?

How many ways can you hang 8 shirts on

hangers in a closet?

The number of permutations of n items of a

set arranged r items at a time is

𝑛𝑃𝑟 =

𝑛!

𝑛 − 𝑟 !for 0 ≤ 𝑟 ≤ 𝑛

Example: 10𝑃4 =10!

10−4 !=

10!

6!= 5040

Ten students are in a race. First, second,

and third places will win medals. In how

many ways can 10 runners finish first,

second, and third (no ties allowed)

What if in the last example the first three

runners would get to go to the state

championship. Then order wouldn’t matter.

A selection in which order doesn’t matter is

called a combination.

The number of combinations of n items of a set chosen r items at a time is

𝑛𝐶𝑟=𝑛!

𝑟! 𝑛−𝑟 ! 𝑓𝑜𝑟 0 ≤ 𝑟 ≤ 𝑛

Example: 5𝐶3 =5!

3! 5−3 !=

5!

3!∙2!=

120

6∙2= 10

How many committees of 4 could you choose

from 13 people?

An art teacher divides his class into five

groups. Each group submits one sketch.

She will select 3 of the sketches to display.

In how many different ways can she select

the sketches?

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?

Experimental probability of an event:

𝑃 𝑒𝑣𝑒𝑛𝑡 =𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑒𝑣𝑒𝑛𝑡 𝑜𝑐𝑐𝑢𝑟𝑠

𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑖𝑎𝑙𝑠

The probability of an impossible event is 0

(or 0%). The probability of a certain event

is 1 (or 100%). Otherwise, the probability of

an event is an number between 0 and 1 (or a

percent between 0% and 100%).

On Thursday, there were 68 cars in the

teacher’s parking lot. Fourteen of the cars

were SUV’s. What is the experimental

probability that a vehicle picked as random

is an SUV?

What is the experimental probability that

the vehicle is NOT an SUV?

A baseball player got a hit in 20 of his last

50 times at bat. What is the experimental

probability that he will get a hit in his next

at bat?

On a multiple-choice test, each item has 4

choices, but only one choice is correct.

How can you simulate guessing the answers?

What is the probability that you will pass

the test by guessing at least 6 of 10 answers

correctly?

The set of all possible outcomes to an experiment or activity is call the sample space. When each outcome in a sample space has the same chance of occurring, the outcomes are equally likely outcomes.

When outcomes are equally likely, you can calculate the theoretical probability of the event.

The theoretical probability of event A is

𝑃 𝐴 =𝑚

𝑛

Sample space: n outcomes

Event A:

m outcomes

What is the theoretical probability of getting

a 5 on one roll of a standard number cube?

What is the theoretical probability of getting

a sum of 5 on one roll of two standard

number cubes?

What is the probability of being dealt exactly

two 7’s in a 5-card hand from a standard 52-

card deck?

What is the probability of being dealt 3 5’s

and 2 aces?

What is the probability that a dart thrown at

this board lands in the bullseye?

20 in

3in

What is the probability that a

checker thrown on top of this

board lands on a black square?