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Unit 8
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 =
Example:
10𝑃4
𝑛!
for 0 ≤ 𝑟 ≤ 𝑛
𝑛−𝑟 !
=
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
𝑛!
𝑛𝐶𝑟 =𝑟! 𝑛−𝑟 !
Example:
𝑓𝑜𝑟 0 ≤ 𝑟 ≤ 𝑛
5!
5𝐶3 =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:
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑒𝑣𝑒𝑛𝑡 𝑜𝑐𝑐𝑢𝑟𝑠
𝑃 𝑒𝑣𝑒𝑛𝑡 =
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 a 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
𝑚
𝑃 𝐴 =
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 52card 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?
What is the probability a quarterback will
complete his next pass if he has completed
30 of his last 40 passes?
What is the probability a quarterback will
complete his next pass if he has completed
36 of his last 45 passes?
Find the probability of each event when
rolling a standard die.
P(3)
P(2 or 4)
*
To find the probability of two events occurring
together, you have to decide whether one event
occurring affects the other event.
*Dependent Events—the outcome of a second
event depends on what happens first.
*Independent Events—the first outcome does not
affect the second outcome
1. Roll a number cube. Then spin a spinner.
2. Pick one flash card, then another from a stack of 30
flash cards.
3. Draw one marble from a bag, then replace it and draw
another.
4. Draw one marble from a bag, then draw another
without replacing.
If A and B are independent events, then
𝑃 𝐴 𝑎𝑛𝑑 𝐵 = 𝑃(𝐴) ∙ 𝑃(𝐵)
Example: At a picnic there are 10 diet
drinks and 5 regular drinks. There are also 8
bags of barbecue chips and 12 bags of
regular chips. If you grab a drink and bag of
chips, what is the probability that you get a
diet drink and regular chips?
Two events that cannot happen at the same time are called
mutually exclusive.
You roll a standard die. Are these events mutually exclusive?
1.
2.
3.
Rolling a 2 and a 3
Rolling an even number and a multiple of 3
Rolling an even number and a prime number
𝑃 𝐴 𝑜𝑟 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃 𝐴 𝑎𝑛𝑑 𝐵
If A and B are mutually exclusive events, then
𝑃 𝐴 𝑜𝑟 𝐵 = 𝑃 𝐴 + 𝑃(𝐵)
*A student can take one foreign language each term.
About 37% of students take Spanish. About 15% of
students take French. What is the probability that a
student chosen at random is taking Spanish or French?
Suppose you reach into this dish
and select a token. What is the
probability that the token is
round or green?