Probability
What are the chances?
Definition of Probability
All probabilities are between 0 and 1. That means there are always more possible outcomes than successful outcomes.All probabilities are between 0 and 1. That means there are always more possible outcomes than successful outcomes.
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.
We define probability as outcomespossibleof
successhavetowaysof
#
#
This is called the sample space
How many ways can I win?
Every possibility for an event is called an outcome. For instance, if the event is randomly drawing a card, there are 52 outcomes.
CountingTo solve basic probability questions, we will need to find two numbers:
outcomespossibleof
successhavetowaysof
#
#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?
Let’s draw a tree diagram to show the entire sample space?
First Course: E Or F
Second Course: SC
A SC
A
Third Course:P G
P GP G
P GP G
P G
There are 12 possible schedules
Counting
These tree diagrams are great because they show the entire sample space. They can be cumbersome, though.
Ex 2. A family has 3 children. What is the probability that the 2 youngest will be boys?
outcomespossibleof
successhavetowaysof
#
# 1st child BG
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 1/4P(3 kids, 2 youngest are boys) = 2/8 or 1/4
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:
____ ____ ____ Course 1 Course 2 Course 3
=
This works if we multiply the number of outcomes at each stage.
The fundamental counting theorem states: to calculate the sample space of a multi-staged event, multiply the number of outcomes at each stage.
Remember, if you’re drawing blanks, draw blanks.
Finding the Sample Space
Ex 3. What is the sample space for each event?
a. Rolling a dieb. Flipping 3 coinsc. Drawing a cardd. Drawing 2 cardse. 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
52 52e. ___ ___ = x 2704
Ex 4. I have 3 shirts, 6 pants and 4 pairs of shoes. How many (random) outfits can I create?
Ex 4. I have 3 shirts, 6 pants and 4 pairs of shoes. How many (random) outfits can I create?
____ ____ ____3 x 6 x 4 = 72
And or OrIn 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. A six on the first AND a five on the second?b. A three on the first AND a three on the second?c. 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)
P(rolling a 6 and a 5) = ___ ________________
___ ___
outcomespossibleof
successhavetowaysof
#
#
P(rolling a 6 and a 5) = 1/36
And or OrIn 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)
P(rolling a 3 AND a 3) =
outcomespossibleof
successhavetowaysof
#
#
P(rolling a 3 AND a 3) = 1/36
_____________
a. A six on the first AND a five on the second?b. A three on the first AND a three on the second?c. 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.
And or OrIn 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)
P(rolling an even AND an even) =
outcomespossibleof
successhavetowaysof
#
#
P(rolling an even AND an even) = 1/4
_____________
a. A six on the first AND a five on the second?b. A three on the first AND a three on the second?c. 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.
And or OrIn 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}
P(rolling a pair of 3s OR a pair of 1s) =
outcomespossibleof
successhavetowaysof
#
#
_______ _____
P(rolling a pair of 3s OR a pair of 1s) = 1/18
d. A 3 on the first and a 3 on the second OR a 1 on the first and a 1 on the second.
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. Rolling a 6 and drawing a heart?
b. Rolling a 5 and drawing the 7 of clubs?
c. Rolling a 6 or drawing a heart?
d. Rolling an odd number or drawing a queen?
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. Rolling a 6 and drawing a heart?
b. Rolling a 5 and drawing the 7 of clubs?
c. Rolling a 6 or drawing a heart?
d. Rolling an odd number or drawing a queen?
a. P(6 AND Heart) =
outcomespossibleof
successhavetowaysof
#
#
52
13
6
1
4
1
6
1
24
1
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. Rolling a 6 and drawing a heart?
b. Rolling a 5 and drawing the 7 of clubs?
c. Rolling a 6 or drawing a heart?
d. Rolling an odd number or drawing a queen?
outcomespossibleof
successhavetowaysof
#
#
b. P(5 AND 7 of clubs)52
1
6
1
312
1
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. Rolling a 6 and drawing a heart?
b. Rolling a 5 and drawing the 7 of clubs?
c. Rolling a 6 or drawing a heart?
d. Rolling an odd number or drawing a queen?
outcomespossibleof
successhavetowaysof
#
#
c. P(6 OR heart) 52
13
6
1 )6( heartandP
52
13
6
1
52
13
6
1HOWEVER, some of the times that we rolled a six, we would have also drawn a heart. We cannot count these successes twice!
4
1
6
1
4
1
6
1
8
3
P(A or B)=P(A)+P(B)-P(A and B)Let’s 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?
Bull’s-eye?
1
2
3
4
5
6
7
8
9
N Y
N
N Y N
N N
N Y
Y Y
Y N Y N
Y Y
So what is P(buttered or bull’s-eye)?So what is P(buttered or bull’s-eye)?
outcomespossibleof
successhavetowaysof
#
#
9)( bullseyeorbutteredP 9
What a party game! I’m guaranteed to win!
But wait! I’ve count some of my wins twice!
P(A or B)=P(A)+P(B)-P(A and B)Let’s 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?
Bull’s-eye?
1
2
3
4
5
6
7
8
9
N Y
N
N Y N
N N
N Y
Y Y
Y N Y N
Y Y
So what is P(buttered or bull’s-eye)?So what is P(buttered or bull’s-eye)?
outcomespossibleof
successhavetowaysof
#
#
9)( bullseyeorbutteredP 9
So I must subtract 2 from my wins count. This accounts for the buttered AND bullseye
9
7)( bullseyeorbutteredP
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. Rolling a 6 and drawing a heart?
b. Rolling a 5 and drawing the 7 of clubs?
c. Rolling a 6 or drawing a heart?
d. Rolling an odd number or drawing a queen?
outcomespossibleof
successhavetowaysof
#
#
d. P(odd or queen) = P(odd) + P(queen) – P(odd and queen)
52
4
6
3
52
4
6
3
13
1
2
1
13
1
2
1
13
7
P(A or B)=P(A)+P(B)-P(A and B)
Let’s 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!
We can use a Venn diagram to show this clearly.
bluestriped
4 3 5
7
a. P(blue and striped)=?a. P(blue and striped)=?
Since we are pulling only ONCE, we count the successful events.
Since we are pulling only ONCE, we count the successful events.
19
4)( stripedandblueP
P(A or B)=P(A)+P(B)-P(A and B)
Let’s 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!
We can use a Venn diagram to show this clearly.
bluestriped
4 3 5
7
b. P(blue or striped)=?b. P(blue or striped)=?
Since we are pulling only ONCE, we count the successful events.
Since we are pulling only ONCE, we count the successful events.
)()()(
)(
stripedandbluePstripedPblueP
stripedorblueP
19
4
19
9
19
7)( stripedorblueP
19
12)( stripedorblueP
Perms and CombosWhat is the probability of winning the lotto 6-49?
This type of probability question is one where you’re picking a small group from a big group (ie. A small group of 6 numbers, from a big group of 49 numbers).
outcomespossibleof
successhavetowaysof
#
#
So, how many possible outcomes are there?
When I’m drawing blanks, draw blanksWhen I’m drawing blanks, draw blanks
49 48 47 46 45 44x x x x x = 1 x 1010
This type of calculation can be simplified using factorials.
Factorials
6 factorial is 6x5x4x3x2x1 = 720.
It is written as 6!
10! = 10x9x8x7x6x5x4x3x2x1
10! = 3628800
What is !5
!12
12345
123456789101112
!5
!12
39916806789101112
!5
!12
Factorials
So what is 14 x 13 x 12 x 11 in factorial notation?
!10
!14
Is seems to be 14! But it’s missing 10!
Factorials are very useful when we’re 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 3x x x x
This can be written as!2
!7 To simplify this even further, we say
57P
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
)!(
!
rn
nPrn
n = # in the big group
r = # in the small group
6)!23(
!323
P
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.
I’m picking a ‘small’ group of 8 out of a ‘big’ group of 8 and order matters.
40320!0
!8
)!88(
!888
P
Notice that 0! =1
Combos
When selecting a small group from a big group and the order selected is not important, combinations are used.
)!(!
!
rnr
nCrn
n = # in the big group
r = # in the small group
Ex. How many ways can 2 people be picked from a group of 3?
Ex. How many ways can 2 people be picked from a group of 3?
Group of 3 = A, B, C
A B C
B C A C A B
But, AB = BA so really there are only these options:AB or CB or AC
3)1)(12(
123
)!23(!2
!323
C
Combos
)!(!
!
rnr
nCrn
Ex. In a certain poker game, a player is dealt 5 cards. How many different possible hands are there?
2598960)!552(!5
!52552
C
Small group from a big group, when order doesn’t matter: combo
Big group: 52Small group: 5
So what is the probability of getting a royal flush (A,K,Q,J,10 of 1 suit)?
So what is the probability of getting a royal flush (A,K,Q,J,10 of 1 suit)?
outcomespossibleof
successhavetowaysof
#
#
There is one royal flush for every suit so that’s 4 successes.
649740
1
2598960
4)( flushroyalP
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 doesn’t matter so it’s a combo.
peoplechoosetowaysof
womenchoosetowaysofwomenP
6#
6#)6(
615
67)6(C
CwomenP
715
1
5005
7
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)?
Small group of 6 from big group of 15, order doesn’t matter so it’s a combo.
peoplechoosetowaysof
womenANDmenchoosetowaysofwomenANDmenP
6#
24#)24(
615
2748)24(C
CCwomenANDmenP
143
42
5005
1470
5005
2170
Remember, 6 people are chosen, so if exactly 4 are men, 2 must be women.
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
c. P(at most 2 men are chosen)?
peoplechoosetowaysof
womenANDmanchoosetowaysofwomenANDmanP
6#
51#)51(
615
5718)51(C
CCwomenANDmanP
Remember, ‘at most’ means it could be 1 man AND 5 women OR 2 men and 4 women OR no men and 6 women.
peoplechoosetowaysof
womenANDmenchoosetowaysofwomenANDmenP
6#
42#)42(
615
4728)42(C
CCwomenANDmenP
peoplechoosetowaysof
womenANDmenchoosetowaysofwomenANDmenP
6#
60#)60(
615
6708)60(C
CCwomenANDmenP
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
c. P(at least 2 men are chosen)?Remember, ‘at least’ means it could be 1 man AND 5 women OR 2 men and 4 women.
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)=
615
4728
615
5718
615
6708
C
CC
C
CC
C
CC
13
35005
1155
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