BASIC PROBABILITY
Probability – the chance of something (an event) happening
PROBABILITY =
# of successful outcomes
# of possible outcomes
All probability answers must be between 0 and 1 (inclusive)
0
1
event will
event will
not happen
happen
Answers can be in decimal or fraction form. If your answer is
greater than 1 or negative then it is wrong.
Sample space – a list of all possible outcomes,
usually these values are listed in brackets {…}
Example One: Flipping a coin
sample space = { heads, tails}
P(heads) =
1
2
event you
are looking for
or .50
or 50%
Example Two: Rolling a die
(die is singular for dice)
P(4) =
_1_
6
P(5 or even #) = _1_ + _3_ = _4_
6
6
6
“OR” means
to add the probabilities of each
event together
Example Three: Rolling two dice
When rolling two dice, we are usually looking for
the sum of the dice unless otherwise noted.
There are 36 different ways to roll the
sums of 2 through 12 on two dice.
Sample space of rolling a 4
= { (1,3) ; (2,2) ; (3,1) }
P(4) =
3
36
=
1
12
Example four: Selecting Items
In a drawer there are different colored
pencils. There are 3 red, 4 blue, 5
yellow, and 8 orange.
P (red pencil) =
3
20
P (red pencil, then a blue pencil) =
Need to know whether or not you are replacing
the first pencil before we take the second
With
3
20
X
4
20
=
12
400
=
3
100
Example four: Selecting Items
In a drawer there are different colored
pencils. There are 3 red, 4 blue, 5
yellow, and 8 orange.
P (red pencil, then a blue pencil) =
Without
3
X
20
Words like
“AND” and
“THEN”
often mean
to multiply.
4
19
=
12
380
=
3
95
The number of pencils
decreases by one because we
have already taken one out of
the box.
Section 6.1
Counting Techniques
Page 276
Game 1
Selecting ping-pong balls out of the container.
There is a choice of the letters L, I, O, N, S and each
person will choose and if they spell a legal word
when drawn in order, then they win a prize.
What is the sample space?
List the sample space. Circle the choices in the
sample place that form words.
Another way to find the number of items in the
sample space is to use the Multiplication Principle.
Let use the same example of the letters in the word
LIONS.
When you choose the first letter there are 5 letters
in the container and when you choose the second
letter there are four letters in the container.
“AND” will usually mean to multiply in the problem.
5 X 4 = 20
So the sample space has 20 items in it.
Game 2
The numbers 1 – 9 are on a card and you will
choose 2 of them. Turn your card in and a winner is
choose if the numbers marked on their card match
the winning numbers which are chosen at random.
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
9 x 8 = 72 possibilities
There are two possible
winning number
combination since there is
no way to tell the
difference between the
answers 3,2 and 2,3.
P (winning #) =
2
72
=
1
36
Game 3
You place your dollar on one of 6 numbered
squares and roll two dice. For each time your
number appears you receive one dollar.
1
2
3
4
5
6
1
2
3
4
5
6
You place you dollar on the number 4. You roll the
first die, what is the probability that your number
comes up?
1
P(#4) =
6
You now roll the second die and the probability of
rolling a 4 will be the same.
1
6
Section 6.2
Counting Techniques
Page 282
Mutually exclusive – the event is not part of two
different groups
EXAMPLE:
List the even numbers 1 through 20
and
List the multiples of three 1 through 20
Even = {2,4,6,8,10,12,14,16,18,20}
Multiples of three = {3,6,9,12,15,18}
The numbers 6,12, and 18 are NOT
mutually exclusive because they belong to
both groups.
Factorials – the way of multiplying all the integers
from n to 1, it is denoted n!
Example:
5! = 5x4x3x2x1 = 120
We use factorials when finding out how
many possibilities there will be (the
sample space) when we are using ALL of
the choices.
How many ways can you visit all of your four
classes?
4 choices of where to go first
3 choices of where to go second
2 choices of where to go third
1 choice of where to go last
4x3x2x1=24
or
4! = 24
Permutations – a counting a procedure in which
the order matters. We usually use permutations
instead of factorials when we are using only part of
the total number of items given.
Example:
You want to go visit 3 of the 8 teachers you had last
year. How many different ways can you visit those
teachers?
P(8,3)
or
8 nPr 3
=
336
Book notation
Calculator notation
for permutation
for permutation
Section 6.3
Combinations
Page 294
Combinations – a counting a procedure in which
the order does not matters.
If you have three items A, B, C.
Permutations
ABC BAC CBA
Combinations
equals
ABC
ACB BCA CAB
EXAMPLE:
The Lottery has 50 numbers to choose from and
you must pick 5 of them. You do not have to pick
them in any order.
How many different outcomes are there in this
lottery?
50 nCr 5 = 2,118,760
Section 6.4
Probability
Page 304
Addition Principle for Probabilities
P(A or B ) = P(A) + P(B) – P( A and B)
P(A or B ) = P(A) + P(B) {mutually exclusive}
Example:
There are 45 football players, 20 basketball players,
and 10 players are on both teams.
FOOTBALL
BASKETBALL
35
10
10
55
55
55
BOTH
EXAMPLE:
You are rolling one die
Find P( rolling a 4 or an even #) =
=
4
6
-
1
6
1
6
+
=
3
6
3
6
The 4 is in both sets
Find P( rolling a 4 or an odd #) = 1 + 3
6
6
= 4
6
Conditional Probability
Conditional probability questions are done the
exact same way that regular probability question
are done, except the denominator changes
because we are looking at a smaller portion of the
entire sample space.
Example:
A regular deck of cards has 52 cards in it.
Find P(7) =
4
52
The word “from” is often used in
conditional probability
Find P(face cards from the diamonds) =
3
13
Section 6.5
Binomial Probability
Page 320
The basic idea behind this is that events are either
going to happen or they are NOT going happen.
EXAMPLE:
3 question true/false quiz. How many different
outcomes can the quiz have? What is the
probability of each of the outcomes?
T
T
T
F
F
T
F
T
F
F
TTT
TTF
TFT
T
TFF
FTT
F
T
FTF
F
FFT
FFF
What if we want to do this problem for 10 or 20
problems? How big is out tree graph going to get?
Is this a “good” way to solve the problem?
The answer is an overwhelming NO! A quiz with
10
ten questions has 2
= 1024 different outcomes
20
and a 20-question quiz has 2
= 1,048,576
outcomes. I do not think we want to draw those
graphs.
So we need to use the Binomial probability
shortcut.
EXAMPLE:
Same 3 question true/false Quiz using the shortcut
Always same
Zero True :
C(3,0) (1/2)
Combinations of zero
true answers in three
questions
Fractions add up to
1 but do not have
to be the same
0
Add up to 1st number
(1/2)
Probability
the answer
is true (or
event will
happen)
3
Probability
the answer is
false (or
event won’t
happen)
Zero True :
0
3
C(3,0) (1/2)
(1/2)
(3nCr0) (1/2)^0(1/2)^3
This is how it looks in
the calculator
Zero:
One:
Two:
Three:
(3
(3
(3
(3
nCr
nCr
nCr
nCr
0)
1)
2)
3)
(1/2)^0
(1/2)^1
(1/2)^2
(1/2)^3
(1/2)^3
(1/2)^2
(1/2)^1
(1/2)^0
=
=
=
=
.125
.375
.375
.125
EXAMPLE:
You are rolling five dice at the same time. Make a
probability distribution table for rolling a 4. Make
sure you include each possible outcome.
Expectations
Page 324
Expectations give us the average winning losing
amount per play of the game.
You get paid $3 for each time your number appears
and lose $10 for your number not appearing at all.
Take the probability of each outcome times the
payout for each outcome.
Find the expectations.