From Randomness to
Probability
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A random phenomenon is a situation in which we
know what outcomes could happen, but we don’t
know which particular outcome did or will happen.
In general, each occasion upon which we observe a
random phenomenon is called a trial.
At each trial, we note the value of the random
phenomenon, and call it an outcome.
When we combine outcomes, the resulting
combination is an event.
The collection of all possible outcomes is called the
sample space.
Ex: What light (red, yellow, green) will you
encounter when you drive to an intersection?
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Toss a coin.
◦ It’s equally likely to get heads or tails
◦ Sample space?
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Roll a die.
◦ It’s equally likely to get any one of six outcomes
◦ Sample space ?
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The events are not always equally likely.
◦ A skilled basketball player has a better than 50-50 chance
of making a free throw.
◦ What’s the chance that the light will be green as you drive
to an intersection?
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Toss two coins: Sample space ?
Toss three coins: Sample space ?
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The probability of an event A is
# of outcomes in A
P (A) =
# of possible outcomes
When thinking about what happens with
combinations of outcomes, things are simplified
if the individual trials are independent.
◦ This means that the outcome of one trial doesn’t
influence or change the outcome of another.
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For example, coin flips are independent.
How about drawing cards from a deck of 52
cards?
Empirical Probability
 We used a computer to perform two simulations of
tossing a balanced coin 100 times.
 Note that in a large number of tosses, the coin will land
with heads facing up about half the time.
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The Law of Large Numbers (LLN) says that the
long-run relative frequency of repeated
independent events gets closer and closer to
a single value.
This single value is called the probability of
the event.
Because this definition is based on repeatedly
observing the event’s outcome, this definition
of probability is often called empirical
probability.
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In everyday speech, when we express a
degree of uncertainty without basing it on
long-run relative frequencies or mathematical
models, we are stating subjective or personal
probabilities.
Personal probabilities don’t display the kind
of consistency that we will need probabilities
to have, so we’ll stick with formally defined
probabilities.
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The LLN says nothing about short-run
behavior.
Relative frequencies even out only in the long
run, and this long run is really long (infinitely
long, in fact).
The so called Law of Averages (that an
outcome of a random event that hasn’t
occurred in many trials is “due” to occur)
doesn’t exist at all.
Example
When two balanced dice are rolled, what’s the probability
of rolling a sum of 8?
The sum of the dice can be 8 in five ways. The
probability the sum is 8 is 5/36.
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The most common kind of picture to make is
called a Venn diagram.
Example: 120 students are surveyed about
what subjects they like at school. 40 students
like biology, 30 students like chemistry. 25
like both biology and chemistry. How many
students like neither of the subjects?
1. Probability Rule:
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A probability is a number between 0 and 1.
For any event A, 0 ≤ P (A) ≤ 1.
2. Probability Assignment Rule:
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The probability of the set of all possible outcomes
of a trial must be 1. P (S) = 1 (S is sample space.)
3. Complement Rule:
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The set of outcomes that are not in the event A is
called the complement of A, denoted AC.
The probability of an event occurring is 1 minus
the probability that it doesn’t occur:
P(A) = 1 – P(AC)
4. Addition Rule:
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Events that have no outcomes in common (and,
thus, cannot occur together) are called disjoint
(or mutually exclusive).
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For two disjoint events A and B, the probability
that one or the other occurs is the sum of the
probabilities of the two events.
P (A or B) = P (A) + P (B), provided that A and B
are disjoint.
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5. Multiplication Rule:
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For two independent events A and B, the
probability that both A and B occur is the product
of the probabilities of the two events.
P (A and B) = P (A) x P (B),
Two independent events A and B are not disjoint,
provided the two events have probabilities
greater than zero
Examples
1) According to tables provided by the U.S. National
Center for Health Statistics in Vital Statistics of the
United States, a person aged 20 has about an 80%
chance of being alive at age 65. Suppose that three
people aged 20 are selected at random. What’s the
probability that all three people will be alive at age
65?
2) For a sales promotion, the manufacturer places
winning symbols under the caps of 5% of all Pepsi
bottles. You buy a six-pack. What is the probability
that you win something?
Notation alert:
 In this text we use the notation P (A or B)
and P (A and B).
 In other situations, you might see the
following:
◦ P (A  B) instead of P (A or B)
◦ P (A  B) instead of P (A and B)
Examples
1) A quiz consists of 5 multiple-choice questions, each
with 4 possible answer. If you have no clue and
randomly guess answers, what’s the probability that
you get
a) a perfect score?
b) exactly one correct answer?
c) a 0 score?
d) at least one correct answer?
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Page 379 – 383
Problem # 1, 9, 11, 13, 15, 17, 19, 21, 25,
27, 31, 33, 35, 39, 41.
Birthday problem