Probability

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Probability

• What are the chances?

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Definition of Probability
Probability is the likelihood of an event occur.
This event could be randomly selecting the ace of
spades, or randomly selecting a red sock or a
thunderstorm.
Every possibility for an event is called an
outcome. For instance, if the event is randomly
drawing a card, there are 52 outcomes.
We define probability as
How many ways can I win?
This is called the sample space
All probabilities are between 0 and 1. That means
there are always more possible outcomes than
successful outcomes.
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Counting
To solve basic probability questions, we will
need to find two numbers
This may involve a lot of counting. Tree
diagrams and the FUNdamental Counting Theorem
will help.
Ex1 A university student needs to take a
language course, a math course and a science
course. There are 2 language courses available
(English and French), 3 math courses to choose
from (Stats, Calculus and Algebra) and 2 science
courses available (Physics and Geology). How many
possible schedules are there?
In other words, What is the sample space?
Lets draw a tree diagram to show the entire
sample space?
First Course
E
Or F
There are 12 possible schedules
Second Course
S
A
S
A
C
C
P
P
P
G
G
G
Third Course
P
G
P
P
G
G
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Counting
Ex 2. A family has 3 children. What is the
probability that the 2 youngest will be boys?
1st child B G
2nd child B G B
G
3rd child B G B G B G
B G
There are 8 possible families
How many have the 2 youngest as boys?
2 of ways to have success
P(3 kids, 2 youngest are boys) 2/8 or 1/4
These tree diagrams are great because they show
the entire sample space. They can be cumbersome,
though.
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Counting with the FTC
We can see that to count the total possible
outcomes, we look at the outcomes of each stage
From Ex 1
2
3
2
x
x
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The fundamental counting theorem states to
calculate the sample space of a multi-staged
event, multiply the number of outcomes at each
stage.

____ ____ ____ Course 1
Course 2 Course 3
This works if we multiply the number of outcomes
at each stage.
Remember, if youre drawing blanks, draw blanks.
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Finding the Sample Space
Ex 3. What is the sample space for each event?

• Rolling a die
• Flipping 3 coins
• Drawing a card
• Drawing 2 cards
• Drawing 1 card, putting it back, then drawing
  another.

a. There are 6 outcomes.
b. ___ ___ ___
2 2 2
x x
8
c. There are 52 outcomes.
d. ___ ___
52 51
x
2652
Ex 4. I have 3 shirts, 6 pants and 4 pairs of
shoes. How many (random) outfits can I create?
52 52
e. ___ ___
x 2704
3 x 6 x 4 72
____ ____ ____
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And or Or
In Probability, the words and and or are of
huge importance.
And means that BOTH events occur.
Or means that ONE OF the events occur.
Ex. A pair of dice is rolled. What is the
probability of rolling

• A six on the first
• AND a five on the second?
• b. A three on the first
• AND a three on the second?
• An even number on the first AND
• an even on the second?
• d. A 3 on the first and a 3 on the second OR a
  1 on the first and a 1 on the second.

a. To win in this situation I must roll 2
numbers, therefore there are 2 stages (draw
blanks)
1
1
x
___ ___
P(rolling a 6 and a 5)
_____________
6
6
x
___ ___
P(rolling a 6 and a 5) 1/36
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And or Or
In Probability, the words and and or are of
huge importance.
And means that BOTH events occur.
Or means that ONE OF the events occur.
Ex. A pair of dice is rolled. What is the
probability of rolling
b. To win in this situation I must roll 2 numbers
(2 blanks)

• A six on the first
• AND a five on the second?
• b. A three on the first
• AND a three on the second?
• An even number on the first AND
• an even on the second?
• d. A 3 on the first and a 3 on the second OR a
  1 on the first and a 1 on the second.

1
1
x
P(rolling a 3 AND a 3)
_____________
6
6
x
P(rolling a 3 AND a 3) 1/36
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And or Or
In Probability, the words and and or are of
huge importance.
And means that BOTH events occur.
Or means that ONE OF the events occur.
Ex. A pair of dice is rolled. What is the
probability of rolling
c. To win in this situation I must roll 2
numbers (2 blanks)

• A six on the first
• AND a five on the second?
• b. A three on the first
• AND a three on the second?
• An even number on the first AND
• an even on the second?
• d. A 3 on the first and a 3 on the second OR a
  1 on the first and a 1 on the second.

3
3
x
P(rolling an even AND an even)
_____________
6
6
x
P(rolling an even AND an even) 1/4
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And or Or
In Probability, the words and and or are of
huge importance.
And means that BOTH events occur.
Or means that ONE OF the events occur.
Ex. A pair of dice is rolled. What is the
probability of rolling
d. To win in this situation I must roll 2
numbers (2 blanks). I win if I roll a 3 AND a 3
OR if I roll a 1 AND a 1
d. A 3 on the first and a 3 on the second OR a
1 on the first and a 1 on the second.
1
1
1
1
x
x
P(rolling a pair of 3s OR a pair of 1s)
_______ _____

x
x
6
6
6
6
P(rolling a pair of 3s OR a pair of 1s) 1/18
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And or Or
What we have seen is that, FOR A MULTI-STAGED
EVENT, and means multiply and or means
add.
Ex 4. A die is rolled and a card is randomly
drawn from a deck. What is the probability of

1. Rolling a 6 and drawing a heart?
2. Rolling a 5 and drawing the 7 of clubs?
3. Rolling a 6 or drawing a heart?
4. Rolling an odd number or drawing a queen?

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And or Or
What we have seen is that, FOR A MULTI-STAGED
EVENT, and means multiply and or means
add.
Ex 4. A die is rolled and a card is randomly
drawn from a deck. What is the probability of
a. P(6 AND Heart)

1. Rolling a 6 and drawing a heart?
2. Rolling a 5 and drawing the 7 of clubs?
3. Rolling a 6 or drawing a heart?
4. Rolling an odd number or drawing a queen?

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And or Or
What we have seen is that, FOR A MULTI-STAGED
EVENT, and means multiply and or means
add.
Ex 4. A die is rolled and a card is randomly
drawn from a deck. What is the probability of
b. P(5 AND 7 of clubs)

1. Rolling a 6 and drawing a heart?
2. Rolling a 5 and drawing the 7 of clubs?
3. Rolling a 6 or drawing a heart?
4. Rolling an odd number or drawing a queen?

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And or Or
What we have seen is that, FOR A MULTI-STAGED
EVENT, and means multiply and or means
add.
Ex 4. A die is rolled and a card is randomly
drawn from a deck. What is the probability of
c. P(6 OR heart)

1. Rolling a 6 and drawing a heart?
2. Rolling a 5 and drawing the 7 of clubs?
3. Rolling a 6 or drawing a heart?
4. Rolling an odd number or drawing a queen?

HOWEVER, some of the times that we rolled a six,
we would have also drawn a heart. We cannot count
these successes twice!
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P(A or B)P(A)P(B)-P(A and B)
Lets take a closer look. Consider a party where
we dropped a piece of buttered toast and threw a
dart (with our eyes closed). What is the
probability of the toast landed on the buttered
side OR throwing a bull's-eye?
Trial Landed on butter? Bulls-eye?
1
2
3
4
5
6
7
8
9
So what is P(buttered or bulls-eye)?
N
N
Y
N
Y
N
N
N
9
N
Y
What a party game! Im guaranteed to win!
Y
Y
Y
N
Y
N
But wait! Ive count some of my wins twice!
Y
Y
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P(A or B)P(A)P(B)-P(A and B)
Lets take a closer look. Consider a party where
we dropped a piece of buttered toast and threw a
dart (with our eyes closed). What is the
probability of the toast landed on the buttered
side OR throwing a bull's-eye?
Trial Landed on butter? Bulls-eye?
1
2
3
4
5
6
7
8
9
So what is P(buttered or bulls-eye)?
N
N
Y
N
Y
N
N
N
9
N
Y
So I must subtract 2 from my wins count. This
accounts for the buttered AND bullseye
Y
Y
Y
N
Y
N
Y
Y
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And or Or
What we have seen is that, FOR A MULTI-STAGED
EVENT, and means multiply and or means
add.
Ex 4. A die is rolled and a card is randomly
drawn from a deck. What is the probability of

1. Rolling a 6 and drawing a heart?
2. Rolling a 5 and drawing the 7 of clubs?
3. Rolling a 6 or drawing a heart?
4. Rolling an odd number or drawing a queen?

d. P(odd or queen) P(odd) P(queen) P(odd
and queen)
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P(A or B)P(A)P(B)-P(A and B)
Lets take a closer look. Consider an experiment
where we pulled socks from a drawer. 7 socks are
blue, 7 are white and 9 are striped.
There are only 19 socks in the drawer, though.
How is this possible?
4 of the blue socks are striped!
a. P(blue and striped)?
We can use a Venn diagram to show this clearly.
Since we are pulling only ONCE, we count the
successful events.
striped
blue
4
3
5
7
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P(A or B)P(A)P(B)-P(A and B)
Lets take a closer look. Consider an experiment
where we pulled socks from a drawer. 7 socks are
blue, 7 are white and 9 are striped.
There are only 19 socks in the drawer, though.
How is this possible?
4 of the blue socks are striped!
b. P(blue or striped)?
We can use a Venn diagram to show this clearly.
Since we are pulling only ONCE, we count the
successful events.
striped
blue
4
3
5
7
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Perms and Combos
What is the probability of winning the lotto 6-49?
This type of probability question is one where
youre picking a small group from a big group
(ie. A small group of 6 numbers, from a big group
of 49 numbers).
So, how many possible outcomes are there?
When Im drawing blanks, draw blanks
49
48
47
46
45
44
x
x
x
x
x
1 x 1010
This type of calculation can be simplified using
factorials.
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Factorials
6 factorial is 6x5x4x3x2x1 720.
It is written as 6!
10! 10x9x8x7x6x5x4x3x2x1
10! 3628800
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Factorials
So what is 14 x 13 x 12 x 11 in factorial
notation?
Is seems to be 14! But its missing 10!
Factorials are very useful when were picking a
small group from a big group.
Ex. How many ways are there to randomly select 5
positions out of a group 7 people?
Small group (5) from a big group (7)
7
6
5
4
3
x
x
x
x
To simplify this even further, we say
This can be written as
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Perms
When selecting a small group from a big group and
the order selected is important, permutations are
used.
Ex2. How many ways can I pick a president,
vice-president from a group of 3.
Group of 3 A, B, C
Pres A B C
VP B C A C
A B
n in the big group
r in the small group
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Perms
Ex3. a group of 8 books must be arranged on a
shelf. How many possible arrangements are there?
The word arranged means that order counts.
Im picking a small group of 8 out of a big
group of 8 and order matters.
Notice that 0! 1
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Combos
When selecting a small group from a big group and
the order selected is not important, combinations
are used.
n in the big group
r in the small group
Ex. How many ways can 2 people be picked from a
group of 3?
But, AB BA so really there are only these
options AB or CB or AC
Group of 3 A, B, C
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Combos
Ex. In a certain poker game, a player is dealt 5
cards. How many different possible hands are
there?
Small group from a big group, when order doesnt
matter combo
Big group 52 Small group 5
So what is the probability of getting a royal
flush (A,K,Q,J,10 of 1 suit)?
There is one royal flush for every suit so thats
4 successes.
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Perms, Combos and Probability
Ex. A group consists of 7 women and 8 men. A
group of 6 must be chosen for a committee. What
is
a. P(exactly 6 women are chosen)?
Small group of 6 from big group of 15, order
doesnt matter so its a combo.
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Perms, Combos and Probability
Ex. A group consists of 7 women and 8 men. A
group of 6 must be chosen for a committee. What
is
b. P(exactly 4 men are chosen)?
Remember, 6 people are chosen, so if exactly 4
are men, 2 must be women.
Small group of 6 from big group of 15, order
doesnt matter so its a combo.
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Perms, Combos and Probability
Ex. A group consists of 7 women and 8 men. A
group of 6 must be chosen for a committee. What
is
Remember, at most means it could be 1 man AND 5
women OR 2 men and 4 women OR no men and 6 women.
c. P(at most 2 men are chosen)?
OR
OR
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Perms, Combos and Probability
Ex. A group consists of 7 women and 8 men. A
group of 6 must be chosen for a committee. What
is
Remember, at least means it could be 1 man AND
5 women OR 2 men and 4 women.
c. P(at least 2 men are chosen)?
P(0 m AND 6 w OR 1m AND 5w OR 2m AND 4w)
P(0 m AND 6 w OR 1m AND 5w OR 2m AND 4w)