Download 13 Lesson B

 Roll
a die, flip a coin
 Unique
3 letter arrangements of CAT
 Unique
4 digit arrangements of 1, 2, 3, 4
CP Probability & Statistics
SP 2015
n! = n(n-1)(n-2)…(1)
Example: 5! = 5 * 4 * 3 * 2 * 1
If event A has m outcome and event B has n
outcomes then the number of possible
outcomes for
A or B is m + n
A and B is mn
1 If you have 5 pairs of pants and 7 shirts, how many different outfits
can you make?
5
7
 5 7  35
pants shirts
2 6 students are running in a race. How many different race results could
there be with no ties?
6 5
4 3 2 1
 6!  720
1st 2nd 3rd 4th 5th 6th
3 How many different seating arrangement can a teacher make for a class
of 30, if the classroom has 6 rows with 5 desks per row?
30 seats, the teacher has 30 choices for the first seat
then 29 for the second, 28 for the third ....
30 29 28 ... 2 1  30!  2.6525E32  2.6525 1032
 265, 252,859,800, 000, 000, 000, 000, 000, 000, 000
4 A new restaurant is offering a special: “Four Course Meal for $25”.
The special allows diners to choose from 5 appetizers, 2 salads, 4
entrees and 6 desserts. Diners can choose 1 option for each course.
How many different 4 course combinations are available?
5
2
4
6
 5 2 4 6  240
app salads entrees desserts
5 Craving ice cream? The local ice cream shop offers 24 flavors. You
can get your ice cream in a sugar cone, waffle cone, cake cone or bowl.
Then top it off with your choice from 8 yummy toppings. Assuming
you only choose 1 flavor of ice cream and 1 topping, how many
different combinations can you get?
24
4
8
 24 4 8  768
flavor container toppings
6 How would your number of choices for question 6 change if you could
also choose no toppings?
24
4
9
 24 4 9  864
flavor container topping
7,8 7 and 8 are different. We will come back to these
9 How many combinations can you get if you roll a dice numbered 1 – 6
and cube lettered A,B,C,D,E,F?
6
6
 6 6  36
numbers letters
10 How many different combinations of 5 cards can be drawn from a
standard deck of 52 cards? Order does not matter, so this is different.
We will come back to it.
11 How many different combinations are available if you toss a fair coin
and roll a standard 6 sided die? 2
6
coin die
 2 6  12
12 How many unique sandwiches can be made using the following
choices:
Buns: 4 different types or no bun
Patty: Chicken, Beef, Bison, Black Bean
Lettuce: Romaine, Iceberg, none
Tomato: Yes or no
Onion: Sliced, Grilled, none
Cheese: Cheddar, American, Swiss, Provolone, none
5
4
3
2
3
5
buns patty lettuce tomato onion cheese
 5 4 3 2 3 5  1800
13 So you think you have this? What if for number 12 we add choices of
4 different sauces?
Choosing 1 or none should be an easy calculation
5
4
3
2
3
5
5
buns patty lettuce tomato onion cheese sauce
 5 4 3 2 3 5 5  9000
but what if you can choose any combination of the 4 sauces or no
sauce? – this part is different (we will come back to it)
14 Time to exercise: Suzy wants to go jogging. She has 5 tops, 4 pairs of
shorts and 2 pairs of shoes to choose from. How many different outfits
could she wear?
5
4
2
 5 4 2  40
tops shorts shoes
Number of possible arrangements when
choosing r items from a set of n items and
ORDER DOES NOT MATTERS
n!
C

n r
r !(n  r )!
7 For question 6, how many choices would you have if you could choose
2 different flavors of ice cream and 0-8 toppings (remember you have
container choices too!)
C2
4
9
 276 4 9  9936
flavor container topping
assuming 2 different flavors
24
8 A class wants to elect 2 officers from 10 candidates. How many
different combinations can there be?
order does not matter
10 C2  45
10 How many different combinations of 5 cards can be drawn from a
standard deck of 52 cards? Order does not matter, so this is different.
We will come back to it.
order does not matter
52 C5  2,598,960
13 So you think you have this? What if for number 12 we add choices of
4 different sauces?
What if you can choose any combination of the 4 sauces or no sauce? –
this part is different (we will come back to it)
5
4
3
2
3
5
4 C0 4 C1 4 C2 4 C3 4 C4
buns patty lettuce tomato onion cheese
sauce
 5 4 3 2 3 5 (1 4 6 4 1)  172800
15 The junior class has 4 seats to fill on student council. 950 juniors are
eligible to run. How many different combinations could be chosen?
order does not matter
950 C4  3.3724E10
Number of possible arrangements when
choosing r items from a set of n items and
ORDER MATTERS
n!
P

n r
(n  r )!
The number of distinct permutations of n
objects where n1 of the objects are
identical, n2 of the objects are identical, . .
. , nr of the objects are identical is found
by the formula:
n!
P

n r
n1 !n2 !.....nr !
 How
many different arrangements can be
made using all of the letters in
 MISSISSIPPI
M 1
I
4
S 4
P
2
11!
11 P11 
4!4!2!
11 10 9 8 7 6 5 4! 11 10 9 8 7 6 5 11 10 3 1 7 3 5



 34650
2!4!4!
214 3 21
111111
Probability allows us to go from
information about random samples to
information about the population.
We KNOW
what outcomes
COULD happen
We DO NOT KNOW
which outcomes
WILL happen.
Examples:
Experiment
Flip a coin
Roll a standard die
Outcomes
Heads, Tails
1, 2, 3, 4, 5, 6
Sample Space
The collection of all
possible outcomes
Probability of an event A, P(A), is a number
between 0 and 1 that identifies the
likelihood that event A happens.
Example: Rolling a standard die
P(1) = 1/6
P(2) = 1/6
P(3) = 1/6
P(4) = 1/6
P(5) = 1/6
P(6) = 1/6
 Trial
Single attempt (or realization) of a random
phenomenon
 Outcome
The observed result of a trial
 Independence
(informal definition)
2 events are independent if the outcome of 1
does not influence the outcome of the other.
 Event
Collection of outcomes
We typically label events so we can attach
probabilities to them
Notation: bold capital letter: A, B, C, …
Example: Roll a die and get an even
E is 2,4 or 6
 Sample
Space
The collection of all possible outcomes
Example: Roll a die
S = {1,2,3,4,5,6}
Observed probability
gets closer to the
calculated/theoretical
probability
When the outcomes in a sample space are
equally likely to occur then:
number of times A occurs in sample space
P( A) 
number of items in sample space