Transcript 3.1 PowerPoint

Sta220 - Statistics
Mr. Smith
Room 310
Class #7
• Questions about the project?
Where We’ve Been
• Identified the objective of inferential statistics:
to make inferences about a population on the
basis of information in a sample
• Introduced graphical and numerical
descriptive measure for both quantitative and
qualitative data.
Where We’re Going
• Develop probability as a measure of
uncertainty
• Introduce basic rules for finding probabilities
• Use a probability as a measure of reliability for
an inference
• Use a probability in random sampling
Section 3.1
An experiment is an act or process of
observation that leads to a single outcome that
cannot be predicted with certainty.
A sample point is the most basic outcome of an
experiment.
Example: Listing Sample Point for a
Coin-Tossing Experiment
Two coins are tossed, and their up faces are
recorded. List all sample points for this
experiment.
Figure 3.1 Tree diagram for the coin-tossing
experiment
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reserved.
Figure 3.2 Venn diagrams for the three
experiments from Table 3.1
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reserved.
The sample space of an experiment is the
collection of all its sample points.
Table 3.1
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reserved.
The probability of sample point is a number
between 0 and 1 which measures the likelihood
that the outcome will occur when the
experiment is performed.
Probability Notation
• Probabilities can be expressed as a fraction
(always reduced), decimal, or percent.
• P(A) = 0.123 means the probability of
event A is 0.123
• Unless noted, round probabilities to three
decimal places.
Law of Large Numbers states the relative
frequency of the number of times that an
outcome occurs when an experiment is
replicated over and over again approaches the
true probability of the outcome.
Figure 3.3 Proportion of heads in N tosses of a
coin
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Education, Inc.. All rights
reserved.
Procedure
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Education, Inc.. All rights
reserved.
EXAMPLE
A Probability Model
In a bag of peanut M&M milk
chocolate candies, the colors of the
candies can be brown, yellow, red,
blue, orange, or green. Suppose
that a candy is randomly selected
from a bag. The table shows each
color and the probability of
drawing that color. Verify this is a
probability model.
Color
Probability
Brown
0.12
Yellow
0.15
Red
0.12
Blue
0.23
Orange
0.23
Green
0.15
•All probabilities are between 0 and 1, inclusive.
• Because 0.12 + 0.15 + 0.12 + 0.23 + 0.23 + 0.15 = 1,
rule 2 (sum of all probabilities must equal 1) is satisfied.
Example: Die Tossing Experiment
A fair die is tossed, and the up face is observed.
If the face is even, you win $1. Otherwise, you
lose $1. What is the probability that you win?
Example: Die Tossing Experiment
The sample space is S: {1, 2, 3, 4, 5, 6}
Since the die is balanced, we assign a probability
of 1/6 to each sample points in the sample
space.
1 1 1
1
𝑃 𝐴 = + + =
6 6 6
2
Figure 3.5 Die-toss experiment with event A,
observe an even number
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Education, Inc.. All rights
reserved.
An event is a specific collection of sample
points.
Example: Probability of an EventCoin-Tossing Experiment
Consider the experiment of tossing two
unbalanced coins. Because the coins are not
balanced, their outcomes ( H or T) are not
equiprobable. Suppose the correct probabilities
associated with the sample points are given in
the accompanying table. [Note: The necessary
for assigning the probabilities to sample points
are satisfied.] Consider the event
A: {Observe exactly one head}
B: {Observe at least one head}
Calculate the probability of A and the probability
of B.
Sample Point
Probability
HH
4/9
HT
2/9
TH
2/9
TT
1/9
2 2
4
𝑃 𝐴 = 𝑃 𝐻𝑇 + 𝑃 𝑇𝐻 = + =
9 9
9
4
So 𝑃 𝐴 =
9
4 2 2
𝑃 𝐵 = 𝑃 𝐻𝐻 + 𝑃 𝐻𝑇 + 𝑃 𝑇𝐻 = + +
9 9 9
8
=
9
So 𝑃(𝐵) = 8/9
Probability of an Event
The probability of an event A is calculated by
summing the probabilities of the sample points
in the sample space for A.
Steps for Calculating Probabilities of
Events
1. Define the experiment: that is, describe the
process used to make an observation and the
type of observation that will be recorded.
2. List of the sample points.
3. Assign probabilities to the sample points.
4. Determine the collection of sample points
contained in the even of interest.
5. Sum the sample point probabilities to get the
probability of the event.
Combinations Rule
Example: Selecting 5 Movies from
Suppose a movie reviewer for a newspaper
reviews 5 movies each month. This month, the
reviewer has 20 new movies from which to
make the selection. How many different samples
of 5 movies can be selected from the 20?
N = 20 and n = 5
20
20!
20!
=
=
5
5! 20 − 5 ! 5! 15!
20 ∙ 19 ∙ 18 ∙ ⋯ ∙ 3 ∙ 2 ∙ 1
=
5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 15 ∙ 14 ∙ 13 … 3 ∙ 2 ∙ 1
20 ∙ 19 ∙ 18 ∙ 17 ∙ 16
=
5∙4∙3∙2∙1
= 15,504