The probability that it rains is 70%

The probability that it does NOT rain is 30%

Instinct tells us that for any event E,

the probability that E happens + the probability that E does NOT happen = 100%

P(E) + P(E’) = 100% = 100/100 = 1

Flip 3 coins:

2 x 2 x 2 = 8 possible outcomes

H

T

H

T

H

TH

T

H

TH

TH

T

Flip 3 coins:

2 x 2 x 2 = 8 possible outcomes

H

T

H

T

H

TH

T

H

TH

TH

T

The probability that all land tails up = 1/8

The probability thatat least one lands heads up = 7/8

ALL TAILS

NOT ALL TAILS

FLIP 10 COINS

What is the probability that at least one lands heads up?

10

The number of possible outcomes = 2 = 1024

P( all tails ) = 1/1024

P( NOT all tails ) = P(at least one heads up ) = 1023/1024

Roll 4 dice

What is the probability that at least two land on the same number?

Some of the possible outcomes:

2 3 1 65 2 5 14 4 4 31 6 3 53 4 2 11 2 6 2..

Favorable outcomes

6 x 6 x 6 x 6 = 1296 possible outcomes

Roll 4 dice

It would be very difficult to count the number of ways in whichat least two land on the same number.

Instead let’s try to count the number of ways in whichit does NOT happen that at least two are the same ….. That is

They ALL land on DIFFERENT numbers

# ways blue die can land:

6

# ways red die can land:

x 5

# ways yellow die can land:

x 4

# ways green die can land:

x 3 = 360

Roll 4 dice

# ways They ALL land on DIFFERENT numbers

# ways blue die can land:

6

# ways red die can land:

x 5

# ways yellow die can land:

x 4

# ways green die can land:

x 3 = 360

6 x 6 x 6 x 6 = 1296 possible outcomes

P( all different ) = 360/1296

P( NOT all different ) = 1 - 360/1296

P( NOT all different ) = 1 - 360/1296

1296/1296 - 360/1296

936/1296

The probability thatyou get an A in Frenchis 1/10

The probability thatyou get an B in Frenchis 3/10

What is the probability that you get an A OR a B in French?

Instinct tells us to add when we see the word OR

1/10 + 3/10 = 4/10 = P( A OR B )

In some cases, there is more to consider.

shuffle this deck of 10 cards and draw one at random.

What is the probability that it is a white card OR a King ?

P ( white card ) = P( W ) = 5/10

P ( King ) = P( K ) = 2/10

P ( white OR King ) = 6/10

10

2

10

5

10

6

KPWPKWP

P ( white card ) = P( W ) = 5/10

P ( King ) = P( K ) = 2/10

P ( white OR King ) = 6/10

10

1

10

2

10

5

10

6

KWPKPWPKWP

Draw 2 cards without replacement from a standard deck of 52 cards.

What is the probability that both are red OR both are kings?

# possible outcomes = 132612

5152252

C

# ways of drawing 2 red cards = 32512

2526226

C

# ways of drawing 2 kings = 612

3424

C

P ( both red ) = P( R ) = 325/1326

P ( both kings ) = P( K ) = 6/1326

P ( both red ) = P( R ) = 325/1326

P ( both kings ) = P( K ) = 6/1326

Both are red AND both are kings =

only 1 of the 1326 possible hands satisfiesthis condition

KR

)( KRP = 1/1326

P ( both red ) = P( R ) = 325/1326

P ( both kings ) = P( K ) = 6/1326

)( KRP = 1/1326

1326

330

1326

1

1326

6

1326

325

)()()()(

KRPKPRPKRP

There are 10 women in a study.

There are 10 women in a study.

3 of them smoke

S

There are 10 women in a study.

3 of them smoke

S

3 develop heartdisease

H

There are 10 women in a study.

S

3 develop heartdisease

H

The probabilitythat a woman in this sample developsheart disease = P(H)=3/10

HS

The probability thatshe gets heart diseaseIF she smokes =

S

The probability thatshe gets heart diseaseIF she smokes = 2/3

HS

P(H) = 3/10 = 30%

P(H IF S)P(H/S) = 2/3 = 66.7%

Based on this data, we might conclude that a woman is more than twice as likely to develop heart disease IF she smokes.

HS

P(H IF S)P(H/S) = 2/3 = 66.7%

)(

)(

)(

)(

3

2)/(

SP

SHP

Sn

SHnSHP

HS

P(H IF S)P(H/S) = 2/3 = 66.7%

)(

)(

)(

)(

3

2)/(

SP

SHP

Sn

SHnSHP

The probability that B happens IF A happens =

the probability that A happens AND B happens the probability that A happens

)(

)()/(

AP

BAPABP

We rewrite this formula to obtain a formula for measuring the probability that A happens AND B happens

)(

)()/(

AP

BAPABP

Multiply both sides of the equation by P(A):

)(AP )(AP

)()/( BAPABP )(AP

)/()()( ABPAPBAP

)/()()( ABPAPBAP

There is a 40% chance that my car won’t start IF it rains.There is a 70% chance of rain.

What is the probability that it rains AND my car won’t start.

R C

)/()()( RCPRPCRP (.70) (.40)

= .28

A jar contains2 red and3 blue marbles.

Draw two without replacement.

What is the probabilitythey are both blue?

10

3

#

#

25

23 C

C

possible

favorable

P(both blue) =P(first is blue AND second is blue)

)/()()( 12121 BBPBPBBP

5

3

4

2

10

3

20

6

)/()()(

)(

)()/(

)()()()(

1)()(

ABPAPBAP

AP

BAPABP

BAPBPAPBAP

EPEP

NOT

OR

IF

AND