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Prob & Stat – 300
NAME____________________________________
Mr. Coppock
Conditional Probability and The Addition Rule
Conditional Probability
The probability we assign to an event can change if we know that some other even has occurred.
This idea is the key to many applications of probability.
Notation: The probability of B given that A occurred is P(B│A) .
Conditional probability is generally solved intuitively.
Examples:
Refer to the table to answer the following questions. 3200 people were surveyed.
Ice cream preference
Likes
Likes Vanilla
Likes
Total
Chocolate Ice
Ice Cream
Strawberry Ice
Cream
Cream
Under 30
604
366
322
1,292
30-51
404
424
286
1,114
Age
Over 51
336
72
386
794
Total
1,344
862
994
3200
a.) Find the probability that a randomly selected individual is over 51.
b.) Find the probability that a randomly selected individual likes vanilla ice cream.
c.) What is the probability you choose someone who likes strawberry ice cream given that you
selected someone under 30?
d.) What is the probability you selected someone under 30 given that you selected someone who
likes strawberry ice cream?
e.) Suppose you randomly choose an individual who is between 30 and 51, what is the
probability the person you chose like chocolate ice cream?
Let’s take a closer look at the multiplication rule.
5. Two events A and B are independent if knowing that one occurs does not change the
probability that the other occurs. If A and B are independent then the probability of A and
B is the product of their individual probabilities.
In symbols:
If A and B are independent, P(A and B) = P(A) • P(B)
Note: This rule is known as the multiplication rule for independent events. Independence cannot
be pictured by a Venn diagram, because it involves the probabilities of the events rather than just
the outcomes that make up the events.
So what happens if the events are not independent (dependent)?
This leads us to the General Multiplication Rule for Any Two Events
The joint probability that events A and B both happen can be found by
P(A and B) = P(A) • P(B│A)
Equivalently
P(A
B) = P(A) • P(B│A)
Here P(B│A) is the conditional probability that B occurs, given the
Information that A occurs.
Examples
1.) Matt McGillicuddy is driving on the highway when he notices 2 gas stations up the road.
The probability that the first gas station is open is 0.57. The probability of the second being open
is 0.42. What is the probability that both gas stations are closed? (assume independence)
2.) In Texas hold ’em, you are dealt 2 cards. A great starting hand would be 2 aces. What is the
probability you are dealt 2 aces?
The Addition Rule
Example
Name
Taylor
Hari
JG
Danny
Emma
Mike
Anna
Cooper
Brendan
Stephen
Anne
Jessie
Patrick
Drew
Julia
Peter
Emily
Sonia
Madison
Perri
Tori
Courtney
Sex
m
m
m
m
f
m
f
m
m
m
f
f
m
m
f
m
f
f
f
f
f
f
Handedness
Eye color
A student from this class is chosen at random. Find the following probabilities
a.) P(Boy)
b.) P(Girl)
c.) P(Boy or Girl)
d.) P(Brown Eyes)
e.) P(Boy or Brown Eyes)
f.) P(Lefty)
g.) P(Girl or Lefty)
In order to understand the addition rule there are a few things we need to get to know:

A compound event is any event combining 2 or more simple events
Ex: Random selection of a female or a person under 60 years old.

Events that are mutually exclusive (or disjoint) are events that CANNOT occur
simultaneously.
Ex: Selecting a person with blue eyes, selecting a person with brown eyes.

Events that are NOT mutually exclusive are events that CAN occur simultaneously
Ex: Selecting a person with blue eyes, selecting a person who is left-handed
So what you need to decide when you do a compound probability problem is decide whether or
not the events are mutually exclusive.
Notation:
The addition rule if the events are “mutually exclusive” (they can’t happen simultaneously)
P(A or B) = P(A) + P(B)
P(A  B) = P(A) + P(B)
The addition rule if the events are not “mutually exclusive” (they can happen simultaneously)
P(A or B) = P(A) + P(B) – P(A and B)
P(A  B) = P(A) + P(B) – P (A  B)
Example #1 Given a standard deck of cards that has been well shuffled. Suppose you chose a
card at random.
a.) P(a club)?
b.) P(a 5)?
c.) P(5 or a club)?
d.) P(a 7 or a jack)?
e.) P(a face card or an ace)?
f.) P(a face card or a red card)?