AP Statistics Section 6.2 A
Probability Models
If you tossed a coin 5 times, would
you be surprised if you got heads
exactly 1 time? If you tossed a coin
50 times, would you be surprised if
you got heads exactly 1 time?
The difference in your answers is
explained by the idea that chance
behavior is ____________
unpredictable in the
short run but has a
__________________
predictable behavior in the long
run.
The word random in statistics is
not a synonym for “haphazard” but
a description of a kind of _______
pattern
that emerges only in the
_________.
long run
We often encounter the unpredictable side of
randomness in our everyday experience, but we
rarely see enough repetitions of the same
random phenomenon to observe the long-term
regularity that probability describes.
In the very long run, the proportion of heads is
0.5. This is the intuitive side of probability. A
probability of 0.5 means “occurs half the time in
a very large number of trials.”
We call a phenomenon random if
individual outcomes are uncertain
but there is, nonetheless, a regular
distribution of outcomes in a large
number of repetitions.
The probability of any outcome of
a random phenomenon is the
proportion of times the outcome
would occur in a large number of
repetitions.
In other words, probability is long
term relative frequency

# occurances
# trials
The idea of probability is __________.
empirical
That is, it is based on ____________
observation
rather than theorizing. Probability
describes what happens in very many
trials, and we must actually observe
many trials to pin down a probability.
The sample space (S) of a random
phenomenon is the set of all
possible outcomes.
An event is any outcome or set of
outcomes of a random
phenomenon.
An event is always a subset of the
sample space.
A probability model is a mathematical
description of a random phenomenon
consisting of two parts:
1. The sample space.
2. A way of assigning probabilities to
events.
Example: What is the sample space for ….
a. rolling two dice? (Think of rolling a red
die and green die.)
1 - 1,
1 - 2,

1 - 3,
s
1 - 4,
1 - 5,

1 - 6,
2 - 1, 3 - 1, 4 - 1, 5 - 1, 6 - 1 

2 - 2, 3 - 2, 4 - 2, 5 - 2, 6 - 2
2 - 3, 3 - 3, 4 - 3, 5 - 3, 6 - 3 

2 - 4, 3 - 4, 4 - 4, 5 - 4, 6 - 4
2 - 5, 3 - 5, 4 - 5, 5 - 5, 6 - 5 

2 - 6, 3 - 6, 4 - 6, 5 - 6, 6 - 6 
Example: What is the sample space for ….
b) tossing 4 coins?
HHHH
T HHH

HT HH
s
HHT H
HHHT


T T HH HT T T

T HT H T HT T
T HHT T T HT

HT T H T T T H
HT HT T T T T


HHT T
Example: What is the sample space for ….
c. flipping a coin followed by throwing a die
T - 1
T - 2

T - 3
s
T - 4
T - 5

T - 6
H -1 

H - 2
H - 3 

H - 4
H -5

H - 6 
Being able to determine the
outcomes in a sample space is
critical to determining
probabilities. One way to do this is
by using a ___________.
tree diagram The tree
diagram part c above looks like:
T
T
T
T
T
T
The previous example illustrates
the Multiplication Principle: If you
can do one task in n1 ways and a
second task in n2 ways, then both
n1  n2
tasks can be done in ______
number of ways.
If you are drawing playing cards twice from a
standard deck of 52 cards, then the second draw
depends on what you do with the first card.
If you put the first card back in the deck before
drawing the second card, that is sampling
________________.
with replacement
If you do not put the first card back in the deck
before drawing the second card, you are
sampling __________________.
without replacement
Example: How many 3 letter
“words” are there if letters can be
repeated?
26 26 26  17,576
Example: How many 3 letter
“words” are there if letters cannot
be repeated?
26 25 24  15,600