Download 3.2 Conditional Probability & the Multiplication Rule

Conditional Probability & the
Multiplication Rule
Conditional Probability
• Is the probability of an event occurring
given that another event has already
occurred. The conditional probability of
event B occurring, given that event A has
occurred, is denoted by P(B|A) and is read
as “probability of B, given A.”
Finding conditional probabilities
Gene
Gene
Total
Present not
present
High
IQ
33
19
52
Normal
IQ
39
11
50
Total
72
30
102
• The table shows the results of a
study in which researchers
examined a child’s IQ and the
presence of a specific gene in
the child. Find the probability
that the child has a high IQ,
given that the child has the
gene.
Solution: There are 72 children
who have the gene. So, the
sample space consists of these
72 children. Of these, 33 have
high IQ. So, P(B|A) = 33/72 ≈
.458
Independent and Dependent Events
• The question of the interdependence of two
or more events is important to researchers in
fields such as marketing, medicine, and
psychology. You can use conditional
probabilities to determine whether events
are independent or dependent.
Definition
• Two events are independent if the occurrence of
one of the events does NOT affect the probability
of the occurrence of the other event. Two events
A and B are independent if:
P(B|A) = P(B) or if P(A|B) = P(A)
Events that are not independent are dependent.
Classifying events as independent or
dependent
• Decide whether the events are independent or
dependent.
1. Selecting a king from a standard deck (A), not
replacing it and then selecting a queen from the
deck (B).
Solution: P(B|A) = 4/51 and P(B) = 4/52. The
occurrence of A changes the probability of the
occurrence of B, so the events are dependent.
Classifying events as independent or
dependent
• Decide whether the events are independent or
dependent.
2. Tossing a coin and getting a head (A), and then
rolling a six-sided die and obtaining a 6 (B).
Solution: P(B|A) = 1/6 and P(B) = 1/6. The
occurrence of A does not change the probability of
the occurrence of B, so the events are independent.
Classifying events as independent or
dependent
• Decide whether the events are independent or
dependent.
3. Practicing the piano (A), and then becoming a
concert pianist (B).
Solution: If you practice the piano, the chances of
becoming a concert pianist are greatly increased,
so these events are dependent.
The Multiplication Rule
• To find the probability of two events occurring in a
sequence, you can use the multiplication rule.
The probability that two events A and B will occur in
sequence is
P(A and B) = P(A) ● P(B)
If events A and B are independent, then the rule can be
simplified to P(A and B) = P(A) ● P(B). This simplified
rule can be extended for any number of events.
Using the multiplication rule to find
probabilities
Two cards are selected without replacement, from a
standard deck. Find the probability of selecting a
king and then selecting a queen.
Solution: Because the first card is not replaced, the
events are dependent.
P(K and Q) = P(K) ● P(Q|K)
4 4
16
  
 0.006
52 51 2652
So the probability of selecting a king and then a
queen is about .0006
Using the multiplication rule to find
probabilities
A coin is tossed and a die is rolled. Find the
probability of getting a head and then rolling a 6.
Solution: The events are independent
1 1 1
P(H and 6) = P(H) ● P(6)     0.083
2 6 12
So the probability of tossing a head and then rolling
a 6 is about .0083
Note:
• To determine if A and B are independent,
calculate P(B) and P(B|A). If the values are
equal, then the events are independent and
the multiplication rule is simplified.
Using the multiplication rule to find
probabilities
• A coin is tossed and a die is rolled. Find the
probability of getting a head and then
rolling a 2.
P(H) = ½. Whether or not the coin is a
head, P(2) = 1/6—The events are
independent.
1 1 1
P( Hand 2)  P( H )  P(2)   
 0.083
2 6 12
So, the probability of tossing a head and then rolling a
two is about .083.
Using the multiplication rule to find
probabilities
• The probability that a salmon swims successfully
through a dam is .85. Find the probability that 3
salmon swim successfully through the dam.
The probability that each salmon is successful is
.85. One salmon’s chance of success is
independent of the others.
P(3salmonaresuccessful )  (.85)(. 85)(. 85)  .614
So, the probability that all 3 are successful is about
.614.
Using the multiplication rule to find
probabilities
• Find the probability that none of the salmon
is successful.
P(3salmonareNOTsuccessf ul )  (.15)(.15)(.15)  .003
So, the probability that none of the 3 are successful is
about .003.
Using the multiplication rule to find
probabilities
• Find the probability that at least one of the
salmon is successful in swimming through
the dam.
P(atleast1issuccessf ul )  1  P( Nonearesuccessful
 1  .003  .997
So, the probability that at least one of the 3 are
successful is about .997.