probability what are the chances?. definition of probability all probabilities are between 0 and 1....

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





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

_____________








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

_____________







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







a. P(6 AND Heart) =

outcomespossibleof

successhavetowaysof

#

#

52

13

6

1

4

1

6

1

24

1

And or Or







outcomespossibleof

successhavetowaysof

#

#

b. P(5 AND 7 of clubs)52

1

6

1

312

1

And or Or







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







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.


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)=?



)()()(

)(

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


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.


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.


c. P(at most 2 men are chosen)?


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.



6#

42#)42(

615

4728)42(C

CCwomenANDmenP



6#

60#)60(

615

6708)60(C

CCwomenANDmenP


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

probability what are the chances?. definition of probability all probabilities are between 0 and 1....

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