Download P(A given B)

Section 4 – 2C:
Finding the Conditional Probability P(B | R)
Independent Events
P (A AND B )
A bag contains several marbles. You randomly draw one marble from the bag, record itʼs color, and
then you REPLACE the first marble drawn. You then randomly draw a second marble from the
bag and record itʼs color. If the the first marble drawn from the bag is replaced before the second
marble is drawn then the two events are independent events. The sample space for the first event
and the sample space for the second event are the same
If A and B are Independent events then P( A and B ) = P(A) •P(B)
Dependent Events
P (A AND B )
A bag contains several marbles. You randomly draw one marble from the bag, record itʼs color, and
then you DO NOT REPLACE the first marble drawn. You then randomly draw a second
marble from the bag and record itʼs color. When you DO NOT REPLACE the first marble the
contents of the bag changes after the first draw. One of the marbles has been taken out in Event
A and NOT Replaced. If you are given the outcome of Event A then the new sample space for
Event B can be listed and the probability for Event B can be found.
If the outcome of the second event (Event B) depends on the outcome of the first event (Event A)
then we say that the two events are Dependent. The probability of a second marble being selected
is NOT the P(B) used in the independent case. The probability of a specific second marble being
selected depends on what marble was selected in Event A. If the outcome of the second event
(Event B) depends on the outcome of the first event (Event A) then probability of Event B is stated
as “the probability of B given that A has happened”. This is written as P(B | A)
If Event A and Event B are DEPENDENT EVENTS
then
P (A AND B ) = P (A ) • P (B | A )
The original Sample Space is used to find the probability of Event A or P(A). After the outcome of
Event A is given then a new sample space is listed and the the probability of B given that A has
happened is found using the new sample space.
Before we can find the probability of a second marble being drawn we need to know what's in the
bag after the first draw. This will depend on what marble was taken and NOT replaced on the
first draw.
Section 4 – 2C
Page 1 of 7
© 2012 Eitel
Finding the Conditional Probability P(B | A)
GIVEN the outcome of Event A WITHOUT replacement.
Whatʼs In The Bag after Event A
Finding P(B | A)
Example 1
You randomly draw 2 marbles, one at a time WITHOUT replacement.
The two events are dependent.
At the start of the problem a bag contains 18 marbles { 8R , 2 B }
Event A: You draw a Red marble WITHOUT replacement.
The bag now has only 9 marbles left in it and there is one less Red marble.
The bagʼs contents
at the start of
first selection
8 R
2 B
10 marbles
The bagʼs contents
at the start of the
second selection
and
then
without replacement
7R
2 B
9 marbles
Find P(B | R)
P(B | R) means find the probability of drawing a Blue Marble from the SECOND BAG
GIVEN that the first draw was a Red Marble THAT WAS NOT PUT BACK.
The second draw is from the second bag. { 7R , 2B }
P(B | R) on the second draw is
2 possible Blue marbles
2
=
9 total marbles
9
Find P(R | R)
P(R | R) means find the probability of drawing a Red Marble from the SECOND BAG
GIVEN that the first draw was a Red Marble THAT WAS NOT PUT BACK.
The second draw is from the second bag. { 7R , 2B }
P(R | R) on the second draw is
Section 4 – 2C
7 possible Red marbles
7
=
9 total marbles
10
Page 2 of 7
© 2012 Eitel
Example 2
You randomly draw 2 marbles, one at a time WITHOUT replacement.
The two events are dependent.
At the start of the problem the a contains 15 marbles { 2R , 3 G , 10 B }
Event A: You draw a Blue marble WITHOUT replacement.
The second bag now has only 9 marbles left in it and there is one less Blue marble.
The bagʼs contents
at the start of the
first selection
2R
3G
10 B
15 marbles
The bagʼs contents
at the start of the
second selection
and
then
WITHOUT
replacement
2R
3G
9B
14 marbles
Find P(R | B)
P(R | B) means find the probability of drawing a Red Marble from the SECOND BAG
GIVEN that the first draw was a Blue Marble THAT WAS NOT PUT BACK.
The second draw is from the second bag. { 2R , 3G, 9 B }
P(R | B) on the second draw is
2 possible Red marbles
2
=
14 total marbles
14
Find P(G | B)
P(G| B) means find the probability of drawing a Green Marble from the SECOND BAG
GIVEN that the first draw was a Blue Marble THAT WAS NOT PUT BACK.
The second draw is from the second bag. { 2R , 3G, 9 B }
P( G | B ) on the second draw is
3 possible Green marbles
3
=
14 total marbles
14
Find P(B | B)
P(B | B) means find the probability of drawing a Blue Marble from the SECOND BAG
GIVEN that the first draw was a Blue Marble THAT WAS NOT PUT BACK.
The second draw is from the second bag. { 2R , 3G, 9 B }
P( B | B ) on the second draw is
Section 4 – 2C
9 possible Blue marbles
9
=
14 total marbles
14
Page 3 of 7
© 2012 Eitel
Example 3
You randomly draw 3 marbles, one at a time WITHOUT replacement.
The 3 events are dependent.
At the start of the problem the a contains 10 marbles { 2R , 3 G , 10 B }
Event A: You draw a Blue marble WITHOUT replacement.
The second bag now has only 14 marbles left in it and there is one less Blue marble.
Event B: You draw a Blue marble WITHOUT replacement.
The third bag now has only 13 marbles left in it and there is one less Blue marble.
The bagʼs contents
at the start of the
first selection
2R
3G
10 B
15 marbles
and
then
WITHOUT
replacement
The bagʼs contents
at the start of the
second selection
2 R
3G
9B
14 marbles
and
then
WITHOUT
replacement
The bagʼs contents
at the start of the
third selection
2R
3G
8B
13 marbles
Find P(R | 2B)
P(R | 2B) means find the probability of drawing a Red Marble from the THIRD BAG
GIVEN that the first 2 draws were Blue Marbles THAT WAS NOT PUT BACK.
The third draw is from the third bag. { 2 R , 3G, 8 B }
P(R | 2B) on the third draw is
2 possible Red marbles
2
=
13 total marbles
13
Find P(G | 2B)
P(G | 2B) means find the probability of drawing a Green Marble from the THIRD BAG
GIVEN that the first 2 draws were Blue Marbles THAT WAS NOT PUT BACK.
The third draw is from the third bag. { 2R , 3G, 8 B }
P( G | 2B ) on the third draw is
3 possible Green marbles
3
=
13 total marbles
13
Find P(B | 2B)
P(R | 2B) means find the probability of drawing a Blue Marble from the THIRD BAG
GIVEN that the first 2 draws were Blue Marbles THAT WAS NOT PUT BACK.
The third draw is from the third bag. { 2R , 3G, 8 B }
P( B | 2B ) on the third draw is
Section 4 – 2C
8 possible Blue marbles
3
=
13 total marbles
13
Page 4 of 7
© 2012 Eitel
What's the difference between P( G ) , P( G | B ) and P( G | 2B )
WITHOUT REPLACEMENT
Each of the 3 probability questions listed above are asking what is the probability of drawing a Green
Marble from a single bag. The difference is what bag you are choosing the marble from and how
many of each type of marbles are in that bag.
Example 1
You randomly draw 3 marbles, one at a time WITHOUT replacement.
The 3 events are dependent.
The first marble drawn is a Blue marble and the second marble drawn is a blue marble.
The bagʼs contents
at the start of the
first selection
2R
3G
10 B
15 marbles
and
then
WITHOUT
replacement
The bagʼs contents
at the start of the
second selection
2 R
3G
9B
14 marbles
and
then
WITHOUT
replacement
P( G )
P(G) asks what is the probability of drawing a Green Marble
out of the FIRST BAG that contains { 2R , 3G, 10 B }
3 possible green marbles
3
P(G) =
=
15 total marbles
15
P( G | B )
P( G | B ) asks what is the probability of drawing a Green Marble
out of the Second BAG that contains { 2R , 3G, 9 B }
3 possible green marbles
3
P(G) =
=
14 total marbles
14
P( G | 2B )
P( G | 2B ) asks what is the probability of drawing a Green Marble
out of the Third BAG that contains { 2R , 3G, 8 B }
3 possible green marbles
3
P(G) =
=
13 total marbles
13
Section 4 – 2C
Page 5 of 7
The bagʼs contents
at the start of the
third selection
2R
3G
8B
13 marbles
First Bag
2R
3G
10 B
15 marbles
Second Bag
2R
3G
9B
14 marbles
Third Bag
2R
3G
9B
14 marbles
© 2012 Eitel
What's the difference between P( G ) , P( G | B ) and P( G | 2B )
WITHOUT REPLACEMENT
Example 2
You randomly draw 3 marbles, one at a time WITHOUT replacement.
The 3 events are dependent.
The first marble drawn is a red marble and the second marble drawn is a red marble.
Whets the difference between P( G ) , P( G | B ) and P( G | 2B )
The bagʼs contents
at the start of the
first selection
3R
4G
8B
15 marbles
and
then
WITHOUT
replacement
The bagʼs contents
at the start of the
second selection
2 R
4G
8B
14 marbles
and
then
WITHOUT
replacement
P( R )
P(R) asks what is the probability of drawing a Red Marble
out of the FIRST BAG that contains { 3R , 4G, 8 B }
3 possible red marbles
3
P(G) =
=
15 total marbles
15
P( R | R )
P( R | R ) asks what is the probability of drawing a Red Marble
out of the SECOND BAG that contains { 2R , 4G, 8 B }
2 possible red marbles
2
P(G) =
=
14 total marbles
14
P( R | 2R )
P( R | 2R ) asks what is the probability of drawing a Red Marble
out of the Third BAG that contains { 1R , 4G, 8 B }
1 possible red marbles
1
P(G) =
=
13 total marbles
13
Section 4 – 2C
Page 6 of 7
The bagʼs contents
at the start of the
third selection
1R
4G
8B
13 marbles
First Bag
3R
4G
8B
15 marbles
Second Bag
2 R
4G
8B
14 marbles
Third Bag
1R
4G
8B
13 marbles
© 2012 Eitel
What's the difference between P( G ) , P( G | B ) and P( G | 2B )
WITH REPLACEMENT
Example 1
You randomly draw 3 marbles, one at a time WITH replacement.
The 3 events are in dependent.
The first marble drawn is a red marble but it is replaced.
The second marble drawn is a red marble but it is replaced.
Whets the difference between P( R ) , P( R | R ) and P( R | 2R )
The contents of the bag stay the same after each draw
so the probability of drawing a Red marble each time must stay the same.
The bagʼs contents
at the start of the
first selection
3R
4G
8B
15 marbles
and
then
WITH
replacement
The bagʼs contents
at the start of the
second selection
3 R
4G
8B
15 marbles
and
then
WITH
replacement
The bagʼs contents
at the start of the
third selection
3R
4G
8B
15 marbles
P( R )
P(R) asks what is the probability of drawing a Red Marble
out of the FIRST BAG that contains { 3R , 4G, 8 B }
3 possible red marbles
3
P(R) =
=
15 total marbles
15
P( R | R )
3R
4G
8B
15 marbles
Second Bag
P( R | R ) asks what is the probability of drawing a Red Marble
out of the SECOND BAG that contains { 3R , 4G, 8 B }
3 possible red marbles
3
P(R) =
=
15 total marbles
15
P( R | 2R )
P( R | R ) asks what is the probability of drawing a Red Marble
out of the THIRD BAG that contains { 3R , 4G, 8 B }
3 possible red marbles
3
P(R) =
=
15 total marbles
15
P( R ) and P( R | R ) and P( R | 2R ) are all
Section 4 – 2C
First Bag
3 R
4G
8B
15 marbles
Third Bag
3 R
4G
8B
15 marbles
3 possible red marbles
3
=
15 total marbles
15
Page 7 of 7
© 2012 Eitel