Probability Rules

```Probability Rules
Section 4.2
Created by Laura Ralston
4 Basic Probability Rules
• Probability of any event
E is a number (fraction
or decimal) between
and including 0 and 1
• 0 < P(E) < 1
• If an event E cannot
occur, its probability is 0
• P(impossible event) =0
The Complement
• The complement of event E, denoted Ec,
consists of all outcomes in the sample space
that are NOT in event E.
• Example: If the experiment is rolling a die and
event E is rolling a 5, then the complement of
event E is NOT rolling a 5 (1, 2, 3, 4, 6).
• P(E) + P(Ec) = 1 or P(Ec) = 1- P(E)
4 Basic Probability Rules
• If event E is certain to
occur, then the
probability is 1.
• P(definitely happening
event) = 1
• The sum of the
probabilities of all the
outcomes in the sample
space is 1.
 RECALL: Compound Event: any event containing
two or more simple events
 KEY WORD:
OR
 In mathematics, we use an “inclusive” or
 Either one or the other or BOTH
 Example: If I ask for a bologna or cheese sandwich, you
could give me a sandwich with
 Just bologna
 Just cheese
 Bologna and cheese
To determine the probability of a
compound event involving OR, we
must first determine if the two
events are mutually exclusive
Mutually Exclusive
Two events are mutually exclusive if they cannot occur at the same
time (i.e. They have no outcomes in common
Event A and B
Event
A
Event
B
NOT Mutually Exclusive
Mutually Exclusive
Event A
Event B
Examples
 At a large political gathering, you select a person at random
to determine if the person is a Republican or female
 At a large political gathering, you select a person at random
to determine if the person is a Democrat or Independent
 Flip a coin to determine if you get a head or tail
 Roll a die to determine if you get a 4 or 6
 Roll a die to determine if you get 3 or an odd number
 Draw a card from a standard deck to determine if you get a
queen and a heart
 Draw a card from a standard deck to determine if you get a
black card or a diamond
Events are Mutually Exclusive
P(A or B) = P(A) +P(B)
Events are NOT Mutually
Exclusive
P(A or B) = P(A) + P(B)
– P(A and B)
Group
Type
0
A
B
AB
Rh +
39
35
8
4
Rh -
6
5
2
1
If one person is randomly selected, find the probability
P(person is not group A)
P( person has type Rh -)
P(person is group A or type Rh -)
P(person is group A or group B)
P(person is not type Rh +)
P(person is group B or type Rh +)
P(person is group AB or type Rh +)
P(person is group A or group O or type Rh+)
Multiplication Rule for Probability
• RECALL: Compound Event: any event containing
two or more simple events
• KEY WORD:
AND
– Two or more events occur in sequence
– If a coin is tossed and then a die is rolled, you can find
the probability of getting a head on the coin and a 4 on
the die
– If two dice are rolled, you can find the probability of
getting a 6 on the first die and getting a 3 on the second
die
To determine the probability of a
compound event involving AND,
we must first determine if the two
events are independent or
dependent
Independent vs Dependent
Independent
Dependent
• Two events A and B are
independent if the fact that
A occurs does not affect the
probability of B occurring
 When the outcome of the
first event A affects the
outcome or occurrence of
the second event B in such a
way that the probability is
changed, the events are said
to be dependent.
• “With Replacement”
• “With Repetition”
 “Without replacement”
 “Without repetition”
Examples
• Drawing a card from a standard deck and getting a
queen, replacing it, and drawing a second card and
getting a queen
• An drawer contains 3 red socks, 2 blue socks, and 5
white socks. A sock is selected and its color is
noted. A second sock is selected and its color
noted.
• Being a lifeguard and getting a suntan
• Randomly selecting a TV viewer who is watching
“Friends”. Randomly selecting a second TV viewer
who is watching “Friends”
Multiplication Rules
Independent Events
Dependent Events
 P(A and B) = P(A) ∙ P(B)
 P(A and B) = P(A) ∙ P(B|A)
 To find the probability of
two independent events
that occur in sequence, find
the probability of each event
separately and then multiply
 To find the probability of two
dependent events that occur in
sequence, find the probability
of the first event, then “adjust”
the probability of the second
event based on the fact that
the first has occurred, and then
Examples
 Draw two cards with replacement from a standard deck. Find P(Queen
and Queen)
 An drawer contains 3 red socks, 2 blue socks, and 5 white socks. Two
socks are selected without replacement. Find P(White and Red)
 A new computer owner creates a password consisting of two
characters. She randomly selects a letter of the alphabet for the first
character and a digit (0-9) for the second character. What is the
effective deterrent to a hacker?
 In the 108th Congress, the Senate consisted of 51 Republicans, 48
Democrats, and 1 Independent. If a lobbyist for the tobacco industry
randomly selects three different Senators, what is the probability that
they are all Republicans? That is P(Republican and Republican and
Republican).
Conditional Probability
• The probability of the second event B should
take into account the fact that the first event A
• KEY WORDS:
GIVEN THAT
• P(B|A) = (formula)
– Logic is easier
Republican
Democrat
Independent
Male
46
39
1
Female
5
9
0
If we randomly select one Senator, find the
probability
P(Republican given that a male is selected)
P(Male given that a Republican is selected)
P(Female given that an Independent was selected)
P(Democrat or Independent given that a male is
selected)
“At Least One” Probability
• “At least one” is equivalent to “one or more”
• The complement (not) of getting at least one
item of a particular type is that you get NO
items of that type
• To calculate the probability of “at least one” of
something, calculate the probability of NONE,
then subtract the result from 1.
P(at least one) = 1 – P(none)
Examples
 If a couple plans to have 10 children (it could
happen), what is the probability that there will be
at least one girl?
 In acceptance sampling, a sample of items is
randomly selected without replacement and the
entire batch is rejected if there is at least one
defect. The Medtyme Company just manufactured
5000 blood pressure monitors and 4% are
defective. If 3 of the monitors are selected at
random and tested, what is the probability that the
entire batch will be rejected?
Assignment
• p. 171 #1-37 odd
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