Stat 20: Intro to Probability and Statistics
Lecture 12: More Probability
Tessa L. Childers-Day
UC Berkeley
10 July 2014
Today’s Goals
Recap
More Rules and Techniques
Examples
By the end of this lecture...
You will be able to:
Use the theory of equally likely outcomes to carefully
determine the probability of a given event
Determine whether two events are mutually exclusive
Apply the addition rule
Calculate probabilities of more complicated events
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Today’s Goals
Recap
More Rules and Techniques
Examples
Three theories of probability
We think about the situation, and use the appropriate theory
Subjective
Equally Likely Outcomes
Frequency
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Today’s Goals
Recap
More Rules and Techniques
Examples
Properties of Probability
Some things that must be true:
Probabilities are between 0% and 100% (or 0 and 1)
The probabilities of all possible events add to 100% (or 1)
P(something) = 100%− P(opposite thing)
= 1 - P(opposite thing)
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Today’s Goals
Recap
More Rules and Techniques
Examples
Probability Basics
Strategies and formulas for finding probabilities:
Draw a box model, fill it with tickets, draw randomly
Conditional probability P(A|B)
Independence and dependence
Multiplication Rule:
P(A and B) = P(B)×P(A|B) = P(A)×P(B|A)
OR
P(A and B) = P(A)×P(B)
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Today’s Goals
Recap
More Rules and Techniques
Examples
Counting the Ways
It is always an option to list all possible outcomes, see how
many match up
chance =
number of outcomes that match desired event
total number of outcomes
Warning: All possibilities must be listed, not just all combinations.
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Today’s Goals
Recap
More Rules and Techniques
Examples
Counting the Ways (cont.)
Example: Suppose we are flipping a coin 4 times. What is the
probability of getting 2 heads and 2 tails?
List all possibilities:
All 4 Heads
3 Heads, 1 Tail
2 Heads, 2 Tails
1 Head, 3 Tails
All 4 Tails
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Today’s Goals
Recap
More Rules and Techniques
Examples
Counting the Ways (cont.)
Example: Suppose we are flipping a coin 4 times. What is the
probability of getting 2 heads and 2 tails?
List all possibilities:
All 4 Heads: HHHH
(1 way)
3 Heads, 1 Tail: HHHT, HHTH, HTHH, THHH
2 Heads, 2 Tails: HHTT, HTTH, TTHH,
THHT, HTHT, THTH
(6 ways)
1 Head, 3 Tails: HTTT, THTT, TTHT, TTTH
All 4 Tails: TTTT
(4 ways)
(4 ways)
(1 way)
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Today’s Goals
Recap
More Rules and Techniques
Examples
Example: Counting Cards
Find the probability that a single card drawn from a standard deck
is a king or a club.
Let’s list the ways this could happen:
There are 4 kings in the deck
There are 13 clubs in the deck
There are 52 cards in the deck
What is P(K or ♣)?
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Today’s Goals
Recap
More Rules and Techniques
Examples
The Addition Rule
To find the chance of either of two events occurring, add the
chance of the 1st to the chance of the 2nd , and subtract the
chance of both events occurring:
P(A or B) = P(A) + P(B) - P(A and B)
Note that P(A or B) ≥ P(A) and any other parts
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Today’s Goals
Recap
More Rules and Techniques
Examples
The Addition Rule (cont.)
P(A or B) = P(A) + P(B) - P(A and B)
Can use this rule to explain our earlier example: What is the
probability that a single card drawn from a standard deck is a king
or a club?
P(K or ♣) = P(K) + P(♣) - P(K and ♣)
4
13
1
+
−
52 52 52
16
=
= 0.308
52
=
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Today’s Goals
Recap
More Rules and Techniques
Examples
Mutual Exclusivity
2 events are mutually exclusive if one event occurring excludes
the other event from occurring.
P(A and B) = 0.
2 events are not mutually exclusive if one event occurring does
not exclude the other event from occurring.
P(A and B) 6= 0.
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Today’s Goals
Recap
More Rules and Techniques
Examples
Mutual Exclusivity (cont.)
Mutual exclusivity and independence are NOT the same thing!
Independent: P(A|B) = P(A)
Mutually exclusive: P(A and B) = 0
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Today’s Goals
Recap
More Rules and Techniques
Examples
Mutual Exclusivity (cont.)
Mutually Exclusive: P(A and B) = 0
This affects the addition rule:
P(A or B) = P(A) + P(B) - P(A and B)
P(A or B) = P(A) + P(B),
if A and B are mutually exclusive
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Today’s Goals
Recap
More Rules and Techniques
Examples
Mutual Exclusivity (cont.)
P(A or B) = P(A) + P(B),
if A and B are mutually exclusive
Can use this rule to explain our earlier example: Suppose we are
flipping a coin 4 times. What is the probability of getting 2 heads
and 2 tails?
P(2H 2T) = P(HHTT or HTTH or TTHH
or THHT or HTHT or THTH)
= P(HHTT) + P(HTTH) + P(TTHH)
+ P(THHT) + P(HTHT) + P(THTH)
4 4 4 4 4 4
1
1
1
1
1
1
=
+
+
+
+
+
2
2
2
2
2
2
4
1
=6
= 0.375
2
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Today’s Goals
Recap
More Rules and Techniques
Examples
Summary of Rules
The following rules can be applied in calculating probabilities
Complement Rule: P(A) = 1 - P(Not A)
Multiplication Rule: P(A and B) = P(A)×P(B|A)
[and a special case when A and B are independent]
Addition Rule: P(A or B) = P(A) + P(B) - P(A and B)
[and a special case when A and B are mutually exclusive]
Combining these rules allows us to calculate many different
probabilities
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Today’s Goals
Recap
More Rules and Techniques
Examples
Examples
Are the following events mutually exclusive?
1
Select a student in your class,
and he/she has blond hair and
blue eyes
2
Select a student in your
college, and he/she is a
sophomore and a Chemistry
major
3
Select any course in your
college, and it is a calculus
course and an English course
4
Select a registered voter, and
he/she is a Republican and a
Democrat
When rolling a die once you get:
5
An even number, and a
number less than 3
6
A prime number, and an odd
number
7
A number greater than 3, and
a number less than 3
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Today’s Goals
Recap
More Rules and Techniques
Examples
Examples (cont.)
A single card is drawn from a deck. Find the probability of
selecting the following:
1
A 4 or a diamond
2
A club or a diamond
3
A jack or a black card
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Today’s Goals
Recap
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Examples
Examples (cont.)
Three dice are thrown at once. Find the chance that
1
All three dice show 4 spots
2
The third die shows 4 spots, given the first two show 4 spots
3
All three dice show the same number of spots
4
Two or fewer dice show 4 spots
5
The sum of the spots is 5
6
At least one 5 is rolled
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Today’s Goals
Recap
More Rules and Techniques
Examples
Examples (cont.)
In an upper division statistics class there are 18 juniors and 10
seniors. 6 seniors are females, 12 juniors are males. If a student is
selected at random, find the chance of selecting the following:
1
A junior or a female
2
A senior or a female
3
A junior or a senior
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Today’s Goals
Recap
More Rules and Techniques
Examples
Examples (cont.)
An urn contains 6 red balls, 2 green balls, and 2 white balls. Find
the chance of selecting the following:
1
In one draw, a red or white ball
2
In two draws, with replacement, 2 red balls or 2 white balls
3
In draws, without replacement, 2 red balls or 2 white balls
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Today’s Goals
Recap
More Rules and Techniques
Examples
Important Takeaways
When in doubt, count it out
Be careful not to double count
Addition rule
Mutually exclusive vs. non-mutually exclusive events
Next time: Calculating probabilities for independent, binary events
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