AP STATS
Randomness to Probability
Name:_____________________________
Date: _____________ Period 1 2 3 4 5 6 7
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.
Probability

The probability of an event is its long-run relative frequency.
-

For any random phenomenon, each attempt, or trial, generates an
outcome.
–
–

Sometimes we are interested in a combination of outcomes (e.g., a
die is rolled and comes up even). – • When thinking about what
happens with combinations of outcomes, things are simplified if
the individual trials are independent.
The Law of Large Numbers
• The Law of Large Numbers (LLN) says that the long-run relative
frequency of repeated independent events gets closer and closer to the true
relative frequency as the number of trials increases.
–
• The common (mis)understanding of the LLN is that random phenomena
are supposed to compensate some for whatever happened in the past. This
is just not true. For example, when flipping a fair coin, if heads comes up
on each of the first 10 flips, what do you think the chance is that tails will
come up on the next flip?
Probability
• Thanks to the LLN, we know that relative frequencies settle down in the
long run, so we can officially give the name probability to that value.
AP STATS
Randomness to Probability
Name:_____________________________
Date: _____________ Period 1 2 3 4 5 6 7
• Probabilities must be between 0 and 1, inclusive.
Equally Likely Outcomes
• When probability was first studied, a group of French mathematicians
looked at games of chance in which all the possible outcomes were
equally likely.
–
–
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.
However, keep in mind that events are not always equally likely
–
Probability

Two requirements for a probability:
–
–
The rules of Probability
1. Something has to happen rule:
-
AP STATS
Randomness to Probability
Name:_____________________________
Date: _____________ Period 1 2 3 4 5 6 7
2. Complement Rule:
Example 1: List the sample space and tell whether the outcomes are
equally likely. A family has two children; record the genders in order of
birth.
Example 2: The plastic arrow on a spinner for a child's game stops
rotating to point at a color that will determine what happens next.
Determine whether the following probability assignment is legitimate.
Red
0.50
Yellow
0.20
Green
0.15
Blue
0.15
AP STATS
Randomness to Probability
Name:_____________________________
Date: _____________ Period 1 2 3 4 5 6 7
Example 3: A fair coin has come up "heads" 10 times in a row. The
probability that the coin will come up heads on the next flip is...
Example 4: In a survey of American women who were asked to name
their favorite color, 19% said blue, 19% said red, 16% said green, 11%
said yellow, 14% said black, and the rest named another color. If you pick
a survey participant at random, what is the probability that she named
another color?
AP STATS
Randomness to Probability
Name:_____________________________
Date: _____________ Period 1 2 3 4 5 6 7
3. Addition rule for disjoint events
Events that have no outcomes in common (and, thus, cannot occur
together) are called disjoint (or mutually exclusive).
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.
4. Multiplication rule for independent events

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), provided that A and B are
independent.

P(A  B) instead of P(A and B)
AP STATS
Randomness to Probability
Name:_____________________________
Date: _____________ Period 1 2 3 4 5 6 7
“or” – addition rule
“and” – multiplication rule
Example: M & M’s
In 2001 the company that makes m & m’s surveyed nearly every country
in the world to vote on a new color between purple, pink and teal. Purple
was the global winner, however in Japan the percentages were 38% pink,
36% teal, and only 16% purple.
1. What’s the probability that a Japanese M & M’s survey
respondent selected at random preferred either pink or teal?
2. If we pick two respondents at random, what’s the probability
that they both selected purple?
3. If we pick three respondents at random, what's the probability
that at least one preferred purple?
AP STATS
Randomness to Probability
Name:_____________________________
Date: _____________ Period 1 2 3 4 5 6 7
Chapter 15
Terms

For any random phenomenon, each ____________generates an
___________.

An ____________is any set or collection of outcomes.

The collection of all possible outcomes is called the
______________ denoted by S.
Events

When outcomes are equally likely, probabilities for events are easy
to find just by counting.

When the k possible outcomes are equally likely, each has a
probability of 1/k.

For any event A that is made up of equally likely outcomes,
P  A 
count of outcomes in A
count of all possible outcomes
The rule for working with probability rules is to:
The most common _________________ used is a
________________________________.
AP STATS
Randomness to Probability
Name:_____________________________
Date: _____________ Period 1 2 3 4 5 6 7
The General Addition Rule
When two events are disjoint (mutually exclusive) we use:
However when two events are not disjoint we use the General
Addition rule
Conditional Probabilities
When we want the probability of an event from a conditional
distribution, we write _________ and pronounce it “the
probability of B given A.”
A probability that takes into account a given condition is called a
conditional probability.
To find the probability of the event B given the event A, we restrict
our attention to the outcomes in A. We then find the fraction of
those outcomes B that also occurred.
AP STATS
Randomness to Probability
Name:_____________________________
Date: _____________ Period 1 2 3 4 5 6 7
The General Multiplication Rule
When two events are not independent we use the General
Multiplication rule
For any two events A and B
Independence
Independence of two events means that the outcome of one event
does not influence the probability of the other.
With our new notation for conditional probabilities, we can now
formalize this definition:
Events A and B are independent whenever P(B|A) = P(B).
(Equivalently, events A and B are independent whenever
P(A|B) = P(A).)
AP STATS
Randomness to Probability
Name:_____________________________
Date: _____________ Period 1 2 3 4 5 6 7
Drawing without replacement
Sampling without replacement means that once one individual is drawn it
doesn’t go back into the pool.

We often sample without replacement, which doesn’t matter too
much when we are dealing with a large population.

However, when drawing from a small population, we need to take
note and adjust probabilities accordingly.
Drawing without replacement is just another instance of working with
conditional probabilities.
Tree Diagrams

A tree diagram helps us think through conditional probabilities by
showing sequences of events as paths that look like branches of a
tree.

Making a tree diagram for situations with conditional probabilities
is consistent with our “make a picture” mantra.
Example:
AP STATS
Randomness to Probability
Name:_____________________________
Date: _____________ Period 1 2 3 4 5 6 7
Example: Books, TV’s or both
Example: Drawing a card
Example: Politics
Reversing the condition


Suppose we want to know P(A|B), but we know only P(A), P(B),
and P(B|A).
We also know P(A and B), since

 P(A and B) = P(A) x P(B|A)
From this information, we can find P(A|B):
AP STATS
Randomness to Probability
Name:_____________________________
Date: _____________ Period 1 2 3 4 5 6 7
Example: Disease
What can go wrong?

Don’t use a simple probability rule where a general rule is
appropriate:

Don’t assume that two events are independent or disjoint without
checking that they are.

Don’t find probabilities for samples drawn without replacement as
if they had been drawn with replacement.

Don’t reverse conditioning naively.

Don’t confuse “disjoint” with “independent.”
What have we learned?

The probability rules from Chapter 14 only work in special cases—
when events are disjoint or independent.

We now know the General Addition Rule and General
Multiplication Rule.

We also know about conditional probabilities and that reversing the
conditioning can give surprising results.

Venn diagrams, tables, and tree diagrams help organize our
thinking about probabilities.

We now know more about independence—a sound understanding
of independence will be important throughout the rest of this
course.