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 ____________ in the

short run but has a __________________ in the long

run.

unpredictable

predictable behavior

The word random in statistics is not a synonym for “haphazard” but a description of a kind of _______

that emerges only in the _________.

pattern

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

trials#

occurances #

The idea of probability is __________. That is, it is based on

____________ rather than theorizing. Probability describes

what happens in very many trials, and we must actually observe

many trials to pin down a probability.

empiricalobservation

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.)

6-6 6,-5 6,-4 6,-3 6,-2 6,-1

5-6 5,-5 5,-4 5,-3 5,-2 5,-1

4-6 4,-5 4,-4 4,-3 4,-2 4,-1

3-6 3,-5 3,-4 3,-3 3,-2 3,-1

2-6 2,-5 2,-4 2,-3 2,-2 2,-1

1-6 1,-5 1,-4 1,-3 1,-2 1,-1

s

Example: What is the sample space for ….b) tossing 4 coins?

HHTT

TTTT HTHT HHHT

TTTH HTTH HHTH

TTHT THHT HTHH

THTT THTH THHH

HTTT TTHH HHHH

s

Example: What is the sample space for ….c. flipping a coin followed by throwing a die

6-H 6-T

5-H 5-T

4-H 4-T

3-H 3-T

2-H 2-T

1-H 1-T

s

Being able to determine the outcomes in a sample space is

critical to determining probabilities. One way to do this is by using a ___________. The tree diagram part c above looks like:

tree diagram

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

tasks can be done in ______ number of ways.

21 nn

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 ________________.

If you do not put the first card back in the deck before drawing the second card, you are

sampling __________________.

with replacement

without replacement

Example: How many 3 letter “words” are there if letters can be

repeated?

576,17262626

Example: How many 3 letter “words” are there if letters cannot

be repeated?

600,15242526