Download Chapter 10 - Spring-Ford Area School District

Chapter 10
Introducing Probability
Random = individual outcomes are uncertain
but there is a regular distribution of outcomes
in a large number of repetitions
Basic Vocabulary of Probability
Probability = the proportion of times the
outcome would occur in a very long series of
repetitions
◦ We can never observe a probability exactly
◦ Mathematical probability is an idealization based on
imagining what would happen in an indefinitely long
series of trials
◦ The proportion in a small or moderate number of
tosses can be far from the probability. Probability
describes long runs only.
Basic Vocabulary of Probability

Probability of getting heads should be:
_______

Flip 10 times. How many heads did you get?
_______

What is your proportion of heads? _______

Compare to the proportion of heads gotten
for the entire class.
Example: Flipping a coin
250 #10.4
Probability is a measure of how likely an event is to occur.
Match
One of the probabilities that follow with each statement.
0
0.01
0.3
0.6
0.99
1
a. This even is impossible. It can never occur.
b. This event is certain. It will occur on every trial.
c. This event is very unlikely, but it will occur once in a
while in a long sequence of trials.
d. This event will occur more often than not.

Sample space = S = the set of possible
outcomes

Event = an outcome or a set of outcomes of a
random phenomenon.
◦ An event is a subset of the sample space

Probability model = a mathematical
description of a random phenomenon consisting
of two parts: a sample space S and a way of
assigning probabilities to events.

Sample space:

Some events: Rolling a 2, rolling an even
#, rolling a # less than 5

How many outcomes are there if you roll
2 dice?
Example: Rolling a 6-sided die

Here is a chart of the possible outcomes for two dice:

Each individual outcome has a probability of ________.

Give the sample space for the event A = Roll a 5:

Find P(A).
Roll
2
3
4
5
6
7
Prob.
Complete the table
8
9
10
11
12

1. Any probability is a number between 0 and
1.

In symbols: For any event A, .
◦ An event with a probability of 0 never occurs
(=impossible)
◦ An event with a probability of 1 occurs with every
trial (=certain)
◦ An event with probability of 0.5 occurs in half the
trials in the long run.
Probability Rules

2. All possible outcomes together must
have probability 1.

In symbols:
space S, P(S) = 1.

Example: Check this rule by adding the
probabilities on the table above:
Probability Rules
or for any sample

3. If two events have no outcomes in common and can
never occur together ( = disjoint), the probability that
one or the other event occurs is the sum of their individual
probabilities.
◦ In symbols: P(A or B) = P(A) + P(B)
◦ = Addition Rule for Disjoint Events
◦ “or” is the key word meaning to add probabilities together
◦ Disjoint is also called mutually exclusive
◦ An example of disjoint events: Rolling a 2 and rolling an odd #
◦ The addition rule extends to more than 2 events as long as no two
have any outcomes in common.
Probability Rules

4. The probability that an event does not occur is
1 minus the probability that the event does
occur.
◦ In symbols: P(A does not occur) = 1 – P(A)
◦ Called the complement of the event
◦ The probability of an event and its complement always
add to 100% or 1.
◦ Example: Probability of rolling a 5 on 2 dice =
________
◦ Probability of not rolling a 5 on 2 dice =
_________________
Probability Rules

Chapter 10: Introducing Probability
Lesson 2

Discrete = a model with a finite sample
space
◦ List in a table:
 All individual outcomes
 All individual probabilities
◦ Check table values for validity:
 Is 0  P(X)  1 ?
 Is P(x)  1 ?
1.02 is OK)
(allow for round-off error = .98 to
◦ Use the table to find probabilities of events
involving more than one outcome by adding
necessary individual probabilities.
Discrete Probability Models

Faked numbers in tax returns, invoices, or
expense account claims often display
patterns that aren’t present in legitimate
records. Some patterns, like too many round
numbers, are obvious and easily avoided by a
clever crook. Others are more subtle. It is a
striking fact that the first digits of numbers in
legitimate records often follow a model
known as Benford’s Law. Call the first digit of
a randomly chosen record X for short.
Benford’s Law gives this probability model for
X (note the first digit cannot be 0):
Example: (10.7 p 255)
First Digit X
Probability
1
2
3
.301 .176 .125
4
5
6
7
8
9
.097
.079
.067
.058
.051
.046
Find the probability the first digit is 1
Find the probability that the first digit is at least 6.
Find the probability the first digit is greater than 6.
Example cont…

Worksheet...
 Chapter
10: Introducing Probability
Lesson 3

Continuous Probability Models
◦ Continuous = a model that assigns
probabilities as areas under a density curve
 The area under the curve above any range of
values is the probability of an outcome in that
range
 This model is used when assigning individual
probabilities to outcomes is impossible because
there is an infinite number of outcomes
◦ Ex: the actual ounces of water in a glass, the amount
of time it takes to finish a test
◦ (both could have many decimal values)

The total area under the density curve = 1
which corresponds to a total probability of 1

Probabilities are assigned to intervals of
outcomes rather than individual outcomes.
(individual outcomes actually have P(x)=0)

We will use the Normal curve for our
models—we will use the standard normal
table or the TI-83 to get the probabilities
Continuous Probability cont…

On the TI-83:
◦ 2nd – VARS – #2:normalcdf(
,
 For Less than: normalcdf(
,
 greater than: normalcdf(
,
)
)
)
 For between two values: normalcdf(

,
** Round all answers properly to 4
decimal places
)

Heights of young women are normally
distributed with inches and inches. Find
the following probabilities:
◦ A woman is between 68 and 70 inches
◦ A woman is 72 inches or taller.
◦ A woman is 60 inches or shorter
Example

Iowa test vocabulary scores for 7th grade
students are normally distributed with and
◦ Write in terms of X: “the student chosen has
a score of 10 or higher”
◦ Find the probability of the event X.
Example

HW page 269 #49,51,53

Chapter 10: Introducing Probability
Lesson 4

Random Variables
◦ = a variable whose value is a numerical
outcome of a random phenomenon
◦ A probability distribution of a random
variable X tells us what values X can take and
how to assign probabilities to those values.
 Random variables are denoted by capital letters
(usually X and Y)
 Two types: discrete and continuous

Discrete = random variables that have a
finite list of possible outcomes
◦ The distribution of X assigns a probability to
each of these outcomes

X
Ex: possible rolls for 2 standard dice X =
{2,3,4,5,6,7,8,9,10,11,12}
2
3
4
5
6
7
8
9
10
11
12
P(X) 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36

Continuous = random variables that can
take on any value in an interval with
probabilities given as areas under a
density curve
◦ Ex: volume of water in a 16.9 ounce bottle
could be any value
(decimals) depending on filling-error of machines

Do page 261 #16 & 17 below: