Transcript Intro_Conditional_Probability

Conditional Probability
P(A) represents the probability assigned to A, it is the
original or unconditional probability
Sometimes there may be conditions “an event B has
occurred” that affects the probability assigned to A
The conditional probability of an event, A, given that an
event B has happened is denoted P(AB)
P(AB) is read “the probability that A occurs given that
B has occurred”
Conditional Probability, con’t
Given that B has occurred, the relevant sample
space has changed, it is no longer S but consists
only of the outcomes in B
A has occurred if and only if one of the outcomes
in their intersection have occurred
For any events A and B with P(B)>0,
P( A  B)
P( A B) 
P( B)
Conditional Probability, con’t
P( A B) 
P( A  B)
P( B)
A
S
B
Conditional Probability, con’t
We can solve these problems using several
methods:
Use the formula
Venn Diagrams
Frequency Tables
Tree Diagrams
Example-using definition
The probability that event A occurs is .63. The
probability that event B occurs is .45. The
probability that both events occur is .10. Find by
using the definition:
P(AB)
P(BA)
Example-Using Venn Diagram
Suppose that A and B are events with probabilities:
P(A)=1/3, P(B)=1/4, P(AB)=1/10
Find each of the following using a Venn Diagram:
1. P(AB)
2. P(BA)
3. P(ACB)
4. P(BCA)
5. P(ACBC)
Example
Consider the experiment of rolling a fair die twice
All outcomes in S are equally likely
 1,1

1, 2 
1,3
S 
1, 4 
1,5 

1, 6 
 2,1
 2, 2 
 2,3
 2, 4 
 2,5
 2, 6 
 3,1
 3, 2 
 3,3
 3, 4 
 3,5
 3, 6 
 4,1
 4, 2 
 4,3
 4, 4 
 4,5
 4, 6 
 5,1
 5, 2 
 5,3
 5, 4 
 5,5
 5, 6 
 6,1 
 6, 2 
 6,3 
 6, 4 
 6,5 

6,
6
  
Example, Using Table
 1,1

1, 2 
1,3
S 
1, 4 
1,5 

1,6 
 2,1
 2, 2 
 2,3
 2, 4 
 2,5
 2,6 
 3,1
 3, 2 
 3,3
 3, 4 
 3,5
3,6 
 4,1
 4, 2 
 4,3
 4, 4 
 4,5
 4,6 
 5,1
5, 2 
 5,3
5, 4 
 5,5
5,6 
 6,1 
 6, 2 
 6,3 
 6, 4 
 6,5 

6,6
 
Let E=the sum of the
faces is even
Let S2=the second die is
a2
Find
1. P(S2E)
2. P(ES2)
Example, Using Table
One way of doing this is to construct a table of
frequencies:
Event Ac
Event A
A B
Event B
Event B
c
A B
Total A
C
TOTALS
A  B Total B
C
AC  BC
Total Ac
Total Bc
Grand Total
Example, con’t
Let’s try it to find P(S2E) and P(ES2)
Event E
Event Ec
Event S2
Event S2c
Remember: P( E ) 
# of successes
Total # of possible outcomes
TOTALS
Sample Space for Rolling Two Die
 1,1

1,2 
1,3
S 
1,4 
1,5

1,6 
 2,1
 2,2 
 2,3
 2,4 
 2,5
 2,6 
 3,1
3,2 
 3,3
3,4 
 3,5
3,6 
 4,1
 4,2 
 4,3
 4,4 
 4,5
 4,6 
5,1
5,2 
5,3
5,4 
5,5
5,6 
 6,1 

 6,2 
 6,3 

 6,4 
 6,5 

 6,6 
Example
If a fair coin is flipped three times, what is the
probability that it comes up tails at least once
given:
1. No information at all
2. All three coins produce the same side
3. It comes up tails at most once
4. The third flip is heads
Example-Tree Diagram
Three manufacturing plants A, B, and C supply 20, 30 and 50%,
respectively of all shock absorbers used by a certain automobile
manufacturer. Records show that the percentage of defective
items produced by A, B and C is 3, 2 and 1%, respectively. What
is the probability that a randomly chosen shock absorber
installed by the manufacturer will be defective?
What is the probability that the part came from manufacturer A,
given that the part was defective?
What is the probability that the part came from B, given that the
part was not defective?
Independent Events
If two events are independent, the occurrence of one
event has no effect on the probability of the other.
E and F are independent events if
P(EF)=P(E)P(F)
Similarly, P(EF G)=P(E)  P(F)  P(G), etc
Independence of E and F implies that
P(EF)=P(E) and P(F)= P(FE)
If the events are not independent, then they are
dependent.
Independent Events
Consider flipping a coin and recording the
outcome each time.
Are these events independent?
Let Hn=the event that a head comes up on the
nth toss
What is the P(H1 H2)?
What is the probability P(H1 H2 H3)?
Independent Events
You throw two fair die, one is green and the
other is red, and observe the outcomes.
Let A be the event that their sum is 7
Let B be the event that the red die shows an even #
Are these events independent? Explain.
Are these events mutually exclusive? Explain
Independent Events, con’t
You throw two fair die, one green and one red
and observe the numbers. Decide which pairs of
events, A and B, are independent:
1. A: the sum is 5
B: the red die shows a 2
2. A: the sum is 5
B: the sum is less than 4
3. A: the sum is even
B: the red die is even
Conditional Probability and
Independence
If E, F and G are three events, then E and F are
independent, given that G has happened, if
P(EFG)=P(EG)  P(FG)
Likewise, events E, F and G are independent,
given that H has happened, given that G has
happened, if
P(EFGH)=P(EH)  P(FH)  P(GH)
Independence
In the manufacture of light bulbs, filaments, glass
casings and bases are manufactured separately
and then assembled into the final product. Past
records show: 2% of filaments are defective, 3%
of casings are defective and 1% of bases are
defective.
What is the probability that one bulb chosen at
random will have no defects?