Transcript 3.2 Conditional Probability and the Multiplication Rule

3.2 Conditional Probability and the Multiplication Rule
• What are we going to learn about?
– Conditional Probability
– Dependent vs Independent Events
– The Multiplication Rule
3.2 Conditional Probability and the Multiplication Rule
•
Example 1: Suppose a professor has 40 students in her Anatomy and
Physiology class. She collected information about each student’s class
level and political party affiliation. The results are shown in the table.
Freshman
Sophomore
Junior
Senior
Total
Democrat
1
4
5
3
13
Republican
4
8
4
2
18
Other
1
3
3
2
9
Total
6
15
12
7
40
A student is picked at random.
– What is the probability he/she is a republican?
– Suppose we knew the selected student was a senior. Would that change the
probability we found earlier?
3.2 Conditional Probability and the Multiplication Rule
•
Example 2: The table below shows the number of patients admitted for
surgery to Palos Community Hospital and General Hospital along with their
survival status for the month of July.
Palos
General
Total
Survived
120
225
345
Died
80
150
230
Total
200
375
575
Suppose a researcher randomly pulls the file on one of the 575 patients.
– What is the probability that the patient survived their surgery?
– Does knowing which hospital the patient was admitted to change that
probability?
3.2 Conditional Probability and the Multiplication Rule
• Example 3: Suppose we draw two cards from a deck of 52.
– What is the probability that our second card is an ace given that the first
card was an ace and it was not replaced?
• In all three examples, we needed to decide how the extra (or
given) information affected the probability of an event.
• Definition of Conditional Probability
P( B | A ) represents the probability of event B
occurring given that event A has already occurred.
#7 p. 152 (Nursing Majors)
3.2 Conditional Probability and the Multiplication Rule
• Suppose we draw two cards from a deck of 52.
– Find the probability that the second card is a Jack given that the
first card was a Jack and it was not replaced.
P( J2 | J1 ) = 3/51 ≈ 0.059
– Find the probability that the second card is a Jack given that the
first card was a Jack and it was replaced.
P( J2 | J1 ) = 4/52 = 1/13 ≈ 0.077
• If the occurrence of one event influences the chances of
another event occurring, we say the two events are
dependent. Otherwise, the two events are independent.
If events A and B are independent, then we have
P( B | A ) = P( B )
3.2 Conditional Probability and the Multiplication Rule
• We can use the Multiplication Rule to calculate the
probability of consecutive events.
P( A and B ) = P(A) ∙ P( B | A )
• If events A and B are independent, then
P( A and B ) = P(A) ∙ P(B)
Practice using these ideas:
#24 p. 154
#28 p. 155
#32 p. 155