Probability

Any activity with an unpredictable results is called anexperiment.

The results of an experiment are called outcomes and the set of all possible outcomes is the sample space.

Examples: Identify the sample space.

Flip a coin.

Toss a die.

S = {H, T}

S = {1, 2, 3, 4, 5, 6}

The number of outcomes in the sample space S is n(S).

2

6

Experiment Sample Space n(S)

Any subset of the sample space is called an event.

Examples: List the outcomes in each event.

Flip a coin

Toss a die

Toss a die

Draw a card

Flip two coins

The number of outcomes in an event E is n(E).

3

4

4

3

Get heads {H}

Get an even number {2, 4, 6}

Get a 3 or higher {3, 4, 5, 6}

Get an 8 {8, 8, 8, 8}

Get at least one head {HH, HT, TH}

Experiment Event n(E)

1

If n(E) = 0, then P(E) = 0, and the event is impossible.

If E is an event from a sample space S of equally likely

outcomes, the probability of event E is:

If n(E) = n(S), then P(E) = 1 and the event is certain.

Definition: Probability

( )( )

( )

n EP E

n S

Examples: A 6-sided die is rolled once.

The event is impossible.

The event is certain.

P(10) = = 06

0

P(n 10) = = 16

6

P(5) = 6

1

Note that 0 P(E) 1.

Example 1: Two coins are tossed. What is the probability that at least one head comes up?

S = {HH, HT, TH, TT} E = {HH, HT, TH}

( ) 3( )

( ) 4

n EP E

n S

Example 2: A card is drawn at random from a standard deck of 52 cards. What is the probability the card drawn is a face card? S = all 52 cards in the deck n(S) = 52

E = {J, J, J, J, Q, Q, Q, Q, K, K, K, K}

n(E) = 12

13

3

52

12

)(

)()(

Sn

EnEP

Two events A and B are mutually exclusive if they have no outcomes in common, A B = .

Example: When a die is tossed, which events are mutually exclusive?

A: getting an even number C: getting 5 or 6. B: getting an odd number

The Venn diagram shows that only A B = , therefore, only events A and B are mutually exclusive.

C

A

24

6

B1

3

5

JJ

J

BK

Q

A

J

A B

If A and B are events, their union A B, is the event “A or B” consisting of all outcomes in A or in B or in both A and B.

A B = {J, J, J, J, Q, K }

Example: A card is drawn at random from a standard deck of 52 cards.

A: getting a club face card B: getting a jack.

List the outcomes for the event of getting a club face card or getting a jack.

If A and B are events, their intersection, written A B, is the event “A and B” consisting of all outcomes common to both A and B.

Example: A card is drawn at random from a standard deck of 52 cards.

A B = {J}

A: getting a club face card

List the outcomes for the event of getting a club face card and getting a jack.

B: getting a jack.

J

K

Q

A

JJ

J

B

A B

(T, H)

(H, T)

S

(T, T)

(H, H)

A(T, H)

(H, T)

If A is an event, the complement of A, written A , is the event “not A” consisting of all outcomes not in A.

Examples: Two coins are flipped.

= {(H, H), (T, T)} AList the outcomes for the event not getting one head and one tail?

A

Event A is getting one head and one tail.

B

).()()()( BAPBPAPBAP If A and B are events, the probability of “A or B” is:

A

– n( A B)

+ +

A B

= ( + ) + ( + ) –

n(A B) = n( A) + n(B)

).()()( BPAPBAP If A and B are mutually exclusive, then

A B

n(A B) +

A and B are mutually exclusiveA B = 0

= n(A) + n(B)= +

52

26)red( P

52

2)queen red a is card the()queen red( PP

52

28

52

2

52

4

52

26)queen red()queen()red( PPP

)queen red()queen aor red is card the( PP

13

7

Example: A card is drawn at random from a standard deck of 52 cards. What is the probability the card is a red or a queen?

“queen”

Q

Q

52

4)queen( P

7“red”

J

K

Q

5

J

9

8

4

6

10

Q6

7

2

A

9K

4

10

3

5

82

3

Example 2: A card is drawn at random from a standard

deck of 52 cards.What is the probability the card is a spade or a club?

Since these events are mutually exclusive,P(club or spade) = P(club) + P(spade) = .

4

1

4

1

2

1

J 6 7

2

A

9 K

4

10

3

58

Q

“spade” “club”

Q

J 6 7

2

A

9 K

4

10

3

5 8

For example, when flipping two coins, the events “the first coin comes up heads” and “the second coin comes up tails” are independent.

Two events are independent if the fact that one event has occurred has no effect on likelihood of the other event.

( ) ( ) ( )P A B P A P B If A and B are independent events, the probability of “A and B” is:

Example: A card is drawn at random from a standard deck of 52 cards. What is the probability the card is a red queen?

B: the card is a queen

Events A and B are independent.

26 4 1( and ) ( ) ( ) ( )

52 52 26P A B P A B P A P B

A: the card is red

Example: A die is tossed. What is the probability of getting 2 or higher?

It is easier to work with the complementary event

“getting a 1”which has probability .1

6

If A is an event, the probability of the event “not A” is:)(1)( APAP

6

5

6

11)higheror twoa getting()one a gettingnot ( PP