Download the probability that

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
H
H
T
H
H
T
T
H
H
T
T
H
T
T
= 8 possible outcomes
Flip 3 coins:
2 x 2 x 2 = 8 possible outcomes
H
H
T
H
H
NOT ALL TAILS
The probability that
at least one lands heads up = 7/8
T
T
H
T
T
H
T
H
T
ALL TAILS
The probability that all land
tails up = 1/8
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
6
x
6
x
6
x
6
= 1296 possible outcomes
What is the probability that at least two land on the same number?
Some of the possible outcomes:
2
5
4
1
3
1
.
.
3
2
4
6
4
2
1
5
4
3
2
6
6
1
3
5
1
2
Favorable
outcomes
Roll 4 dice
It would be very difficult to count the number of ways in which
at least two land on the same number.
Instead let’s try to count the number of ways in which
it does NOT happen that at least two are the same ….. That is
They ALL land on DIFFERENT numbers
# ways blue # ways red
# ways yellow
die can land: die can land: die can land:
6
x
5
x
4
# ways green
die can land:
x
3
= 360
Roll 4 dice
6
x
6
x
6
x
6
= 1296 possible outcomes
# ways They ALL land on DIFFERENT numbers
# ways blue # ways red
# ways yellow
die can land: die can land: die can land:
6
x
5
x
4
# ways green
die can land:
x
3
= 360
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 that
you get an A in French
is 1/10
The probability that
you get an B in French
is 3/10
What is the probability that you get an A
OR a
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.
B in French?
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
PW  K 
6
10
PW   PK 
5 2

10 10
P ( white card ) = P( W ) = 5/10
P ( King ) = P( K ) = 2/10
P ( white OR King ) = 6/10
PW  K 
6
10


PW   PK   PW  K 
5 2 1
 
10 10 10
Draw 2 cards without replacement from a standard deck of 52 cards.
What is the probability that both are red OR both are kings?
52  51
 1326
# possible outcomes = 52 C2 
2 1
26  25
 325
# ways of drawing 2 red cards = 26 C2 
2 1
# ways of drawing 2 kings =
43
6
4 C2 
2 1
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 =
RK
only 1 of the 1326 possible hands satisfies
this condition
P( R  K )
= 1/1326
P ( both red ) = P( R ) = 325/1326
P ( both kings ) = P( K ) = 6/1326
P ( R  K ) = 1/1326
P( R  K )  P( R)  P( K )  P( R  K )
325
6
1



1326 1326 1326
330

1326
There are 10 women in a study.
S
3 of them smoke
There are 10 women in a study.
S
H
3 of them smoke
There are 10 women in a study.
3 develop heart
disease
S
H
The probability
that a woman in
this sample develops
heart disease = P(H)=
3/10
There are 10 women in a study.
3 develop heart
disease
S
H
The probability that
she gets heart disease
IF she smokes =
S
The probability that
she gets heart disease
IF she smokes = 2/3
S
H
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.
S
H
P(H IF S)
P(H/S) = 2/3 = 66.7%
2 n( H  S ) P ( H  S )
P( H / S )  

3
n( S )
P( S )
S
H
P(H IF S)
P(H/S) = 2/3 = 66.7%
2 n( H  S ) P ( H  S )
P( H / S )  

3
n( S )
P( S )
The probability that B happens IF A happens =
the probability that A happens AND B happens
the probability that A happens
P( B / A) 
P( A  B)
P( A)
We rewrite this formula to obtain a formula for measuring the
probability that A happens AND B happens
Multiply both sides of the equation by P(A):
P( A  B)
P(A) P( B / A)  P(A)
P( A)
P(A) P(B / A) 
P( A  B) 
P( A  B)
P( A)P(B / A)
P( A  B) 
P( A)P(B / A)
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
P(R  C) 
C
P(R)P(C / R)
(.70)
(.40)
= .28
# favorable

# possible
C2
3

10
5 C2
3
P(both blue) =
P(first is blue AND second is blue)
A jar contains
2 red and
3 blue marbles.
Draw two
without replacement.
What is the probability
they are both blue?
P(B1B2 )  P(B1 )P(B2 / B1 )
3
5
2
4
6 3
 
20 10
NOT
OR
IF
AND
P( E )  P( E )  1
P( A  B)  P( A)  P( B)  P( A  B)
P( A  B)
P( B / A) 
P( A)
P( A  B)  P( A) P( B / A)