Download Conditional Prob 2012

Conditional Probability
and General Multiplication Rule
AP Statistics
• Ex. 1) What is the probability of drawing a 7
of clubs, then a 4 of hearts without
replacement?
“given that”
• P(draw 7 clubs) P(draw a 4 of hearts 7 of clubs
was drawn)
• =
1
1

52 51
• Notice how conditional probability “shrinks
the sample space”.
Conditional Probabilities
Here is a contingency table that gives the counts of AP Stats
students by their gender and political views.
(Data are from Fall 2005 Class Survey)
Liberal
Gender Male
17
Female
30
Total
47
Political views
Moderate Conservative Total
29
14
60
24
23
77
53
37 137
P(Female) = 77/137 = 0.562
P(Female and Liberal) = 30/137 = 0.219
What is the probability that a selected student has moderate
political views given that we have selected a female?
3
Conditional Probabilities
(continued)
What is the probability that a selected student has moderate
political views given that we have selected a female?
Liberal
Gender Male
17
Female
30
Total
47
Political views
Moderate Conservative Total
29
14
60
24
23
77
53
37 137
P(Moderate | Female) = 24/77 = 0.311
Conditional probability, P (B|A) – the probability of event
B given event A.
4
Conditional Probabilities
(continued)
Formal Definition:
P  A and B 
P  B | A 
P  B
Example: P(Moderate and Female)
P(Female)
=(24/137) / (77/137)
= 0.175 / 0.562
= 0.311
5
Multiplication Rule for
Independent Events Revisited
Multiplication Rule for Independent events:
P(A and B) = P(A) * P(B)
Independent – the occurrence of one event has no effect on
the probability of the occurrence of another event.
Example: A survey by the American Automobile
Association (AAA) revealed that 60 percent of its
members made airline reservations last year. Two
members are selected at random. What is the probability
both made airline reservations last year?
P(R1 and R2) = P(R1)*P(R2) = (0.6)*(0.6) = .36
6
General Multiplication Rule
Use when events are Dependent.
P(A and B) = P(A) * P(B|A)
For two events A and B, the joint probability that both
events will happen is found by multiplying the probability
event A will happen by the conditional probability of event
B occurring.
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General Multiplication Rule
•P  A and B  P  A   P  B | A 
P(A and B)
P(B|A) 
P(A)
**Also:
P(A and B)
P  A | B 
P(B)
When events are independent:
P(A and B)
P(B|A) 
P(A)
P(A)  P(B)
P(B|A) 
P(A)
P(B|A)=P(B)
P  B|A   P  B
when events A and B are independent!
• When not independent:
P(A and B)
P(B | A) 
P(A)
Ex 1) In a recent study it was
found that the probability that a
randomly selected student is a
girl is .51 and is a girl and plays
sports is .10. If the student is
female, what is the probability
that she plays sports?
P(S and F) .1
P(S|F) 

 .1961
P(F)
.51
Ex 2) The probability that a
randomly selected student plays
sports if (given that) they are male
is .31. What is the probability that
the student is male and plays sports
if the probability that they are male
is .49?
P(S and M)
x
P(S|M) 
.31 
P(M)
.49
x  .1519
Checking for Independence
• The probability of a student in this course getting an
A is 0.35; the probability of being female is 0.60. If
P(A and Female) = 0.15, are the events “getting an A” and “being a female” independent?
• Check to see if P(A)•P(Female) = P(A and Female)
• (0.35)(0.60) ≠0.15
• Thus, the events “getting and A” and “being a female” are not independent.
Homework
• Worksheet