5.4 - STAT3000 Section 5.4 Conditional Probability Suppose...

STAT3000
Section 5.4: Conditional Probability
Suppose you draw 2 cards (with replacement) from a
normal deck of cards. What is the probability you will
draw one spade and then a heart?
A = first card is a spade, B = second card is a heart
P(A and B) =
What if these drawings are done without replacement?
P(A and B) =
Ex : The probability of observing an even number
(event A) on a toss of a fair die is 0.5, where
S = {1, 2, 3, 4, 5, 6} and A = {2, 4, 6}.
Suppose we’re given the information that on a
particular throw of the die the result was a number less
than or equal to 3 (event B), so B = {1, 2, 3}.
Would the probability of observing an even number on
that throw of the die still be equal to 0.5?
66

Ex : A federal agency concerned with rising health costs
and employees without health care, gathered the following
information from several randomly selected companies
nationwide within the same industry.
Company Size Fee-for-Service PPO HMO Totals
Small 1808 1757 1456 5021
Medium 8953 6491 6983 22,382
Large 330,419 241,77
0
233,71
1
805,900
Totals 341,180 250,01
8
242,10
5
833,303
One employee from the 833,303 total employees is to
be chosen at random for further analysis.
A: an employee who chose fee-for-service
B: an employee from a small company
*Over lap so there cannot be disjoint
a.
Find P(B)=5021/833303=.006025
b.
Find P(A and B)= 1808/833303=.00217
c.
Find P(A or B)= 341180/833303+5021/833303-
1808/833303=.4132
d.Find P(A | B)= P(A given that B has occurred)
=P( fee for service given that he works
for small company)=1808/5021=
67

Numerator= P(A and B)
Denominator= P(B)
e. Find P(B | A)
P( small company , given that fee-for-
service)=1808/341180
Numerator= P(A and B)
Denominator= P(A)
Conditional Probability Strategies
1. The Formula
To find the conditional probability that
event A occurs given that event B
occurs, divide the probability that both A
and B occur by the probability that B
occurs.
P(A | B) =
P(A and B) ______
_____________ P(B)_____________
68

Tree Diagram
Revisit the dice example, but use a tree diagram.
P(A | B) =
__________
*
__________
=
If knowing that B occurs gives no additional
information about the probability of A
occurring, then A and B are independent
events.
If P(A | B) = P(A)