Download Know how to find probabilities using theoretical and relative

IQL Chapter 6 – Probability in Statistics
Statistical Reasoning for everyday life, Bennett, Briggs, Triola, 3rd Edition
6.1 The Role of Probability in Statistics: Statistical Significance
FROM SAMPLE TO POPULATION
When the difference between what is observed and what is expected seems unlikely to be explained by
chance alone, we say that the difference is statistically significant.
Definition
A set of measurements or observations in a statistical study is said to be statistically significant if it
is unlikely to have occurred by chance.
QUANTIFYING STATISTICAL SIGNIFICANCE
Quantifying Statistical Significance


If the probability of an observed difference occurring by chance is 0.05 (or 1 in 20) or less,
the difference is statistically significant at the 0.05 level.
If the probability of an observed difference occurring by chance is 0.01 (or 1 in 100) or less,
the difference is statistically significant at the 0.01 level.
6.2 Basics of Probability
LEARNING GOAL
Know how to find probabilities using theoretical and relative frequency methods and understand how to
construct basic probability distributions.
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Definitions
Outcomes are the most basic possible results of observations or experiments.
An event is a collection of one or more outcomes that share a property of interest.
Expressing Probability
The probability of an event, expressed as P (event), is always between 0 and 1 inclusive. A probability of
0 means that the event is impossible and a probability of 1 means that the event is certain.
THEORETICAL PROBABILTY
As long as all outcomes are equally likely, we can use the following procedure to calculate the
theoretical probabilities.
Theoretical Method for Equally Likely Outcomes
Step 1. Count the total number of possible outcomes.
Step 2. Among all the possible outcomes, count the number of ways the event of interest, A, can
occur.
Step 3. Determine the probability, P(A), from
P(A) =
Theoretical Probabilities
Counting Outcomes
Suppose we toss two coins and want to count the total number of outcomes. The toss of the first coin
has two possible outcomes: heads (H) or tails (T). The toss of the second coin also has two possible
outcomes.
The two outcomes for the first coin can occur with either of the two outcomes for the second coin.
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So the total number of outcomes for two tosses is 2 × 2 = 4;
the tree diagram of Figure 6.4a (next slide).
they are HH, HT, TH, and TT, as shown in
Counting Outcomes
Suppose process A has a possible outcomes and process B has b possible outcomes. Assuming the outcomes
of the processes do not affect each other, the number of different outcomes for the two processes
combined is a × b.
This idea extends to any number of processes.
For example, if a third process C has c possible outcomes, the number of possible outcomes for the three
processes combined is a × b × c.
THEORETICAL PROBABILTY
The second way to determine probabilities is to approximate the probability of an event A by making
many observations and counting the number of times event A occurs. This approach is called the
relative (or empirical) method.
Relative Frequency Method
Step 1. Repeat or observe a process many times and
interest,
A, occurs.
count the number of times the event of
Step 2. Estimate P(A) by
P(A) =
EXAMPLE 4 500-Year Flood
Geological records indicate that a river has crested above a particular high flood level four times in the
past 2,000 years. What is the relative frequency probability that the river will crest above the high flood
level next year?
Solution: Based on the data, the probability of the river cresting above this flood level in any single year
is
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Because a flood of this magnitude occurs on average once every 500 years, it is called a “500-year
flood.”
The probability of having a flood of this magnitude in any given year is 1/500, or 0.002.
SUBJECTIVE PROBABILITES
The third method for determining probabilities is to estimate a subjective probability using experience
or intuition.
Three Approaches to Finding Probability
A theoretical probability is based on assuming that all outcomes are equally likely. It is determined by
dividing the number of ways an event can occur by the total number of possible outcomes.
A relative frequency probability is based on observations or experiments. It is the relative frequency
of the event of interest.
A subjective probability is an estimate based on experience or intuition.
PROBABILITY OF AN EVENT NOT OCCURRING
Probability of an Event Not Occurring
If the probability of an event A is P(A), then the probability that event A does not occur is P(not A).
Because the event must either occur or not occur, we can write
P(A) + P(not A) = 1
or
P(not A) = 1 – P(A)
Note: The event not A is called the complement of the event A; the “not” is often designated by a bar,
so Ā means not A.
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PROBABILITY DISTRIBUTION
A probability distribution is a distribution in which the variable of interest is associated with a
probability.
A probability distribution is a table or an equation that links each outcome of a statistical experiment
with its probability of occurrence.
Probability Distribution Prerequisites
To understand probability distributions, it is important to understand variables. Random variables and
some notation.

A variable is a symbol (A, B, x, y, etc.) that can take on any of a specified set of values.

When the value of a variable is the outcome of a statistical experiment, that variable is a
random variable.
Generally, statisticians use a capital letter to represent a random variable and a lower-case letter, to
represent one of its values. For example,

X represents the random variable X.

P(X) represents the probability of X.

P(X = x) refers to the probability that the random variable X is equal to a particular value,
denoted by x. As an example, P(X = 1) refers to the probability that the random variable X is
equal to 1.
Probability Distributions
An example will make clear the relationship between random variables and probability distributions.
Suppose you flip a coin two times. This simple statistical experiment can have four possible outcomes:
HH, HT, TH, and TT. Now, let the variable X represent the number of Heads that result from this
experiment. The variable X can take on the values 0, 1, or 2. In this example, X is a random variable;
because its value is determined by the outcome of a statistical experiment.
A probability distribution is a table or an equation that links each outcome of a statistical experiment
with its probability of occurrence. Consider the coin flip experiment described above. The table below,
which associates each outcome with its probability, is an example of a probability distribution.
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Number of heads
Probability
0
0.25
1
0.50
2
0.25
The above table represents the probability distribution of the random variable X.
http://stattrek.com/probability-distributions/probability-distribution.aspx
Making a Probability Distribution
A probability distribution represents the probabilities of all possible events. Do the following to
make a display of a probability distribution:
Step 1. List all possible outcomes. Use a table or figure if it is helpful.
Step 2. Identify outcomes that represent the same event. Find the probability of each event.
Step 3. Make a table in which one column lists each event and another column lists each
probability. The sum of all the probabilities must be 1.
6.3 Probabilities with Large Numbers
Understand the law of large numbers, use this law to understand and calculate expected values, and
recognize how misunderstanding of the law of large numbers leads to the gambler’s fallacy.
THE LAW OF LARGE NUMBERS
Law of Large Numbers.
The Law of Large Numbers says that in repeated, INDEPENDENT trials with the same probability
p of success in each trial, the percentage of successes is increasingly likely to be close to the
chance of success as the number of trials increases. More precisely, the chance that the
percentage of successes differs from the probability p by more than a fixed positive amount, e
> 0, converges to zero as the number of trials n goes to infinity, for every number e > 0. Note
that in contrast to the difference between the percentage of successes and the probability of
success, the difference between the number of successes and the EXPECTED number of
successes, n×p, tends to grow as n grows. The following tool illustrates the law of large
numbers; the button toggles between displaying the difference between the number of
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successes and the expected number of successes, and the difference between the percentage
of successes and the expected percentage of successes. The tool on this page illustrates the
law of large numbers.
http://www.stat.berkeley.edu/~stark/SticiGui/Text/gloss.htm#l
The idea that large number of events may show some pattern even while individual events are
unpredictable.
The Law of Large Numbers
The law of large numbers (or law of averages) applies to a process for which the probability of
an event A is P(A) and the results of repeated trials do not depend on results of earlier trials
(they are independent).
It states: If the process is repeated through many trials, the proportion of the trials in which
event A occurs will be close to the probability P(A). The larger the number of trials, the closer the
proportion should be to P(A).
EXPECTED VALUE
The mean of a random variable is more commonly referred to as its Expected Value, i.e. the
value you expect to obtain should you carry out some experiment whose outcomes are
represented by the random variable.
The expected value of a random variable X is denoted by
Given that the random variable X is discrete and has a probability distribution f(x), the expected
value of the random variable is given by:
http://www.wyzant.com/Help/Math/Statistics_and_Probability/Expected_Value/
Expected Value
The expected value of a variable is the weighted average of all its possible events. Because it is an
average, we should expect to find the “expected value” only when there are a large number of events,
so that the law of large numbers comes into play.
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Calculating Expected Value
Consider two events, each with its own value and probability. The expected value is
expected value =
(value of event 1) * (probability of event 1)
+ (value of event 2) * (probability of event 2)
This formula can be extended to any number of events by including more terms in the sum.
THE GAMBLERS FALLACY
Definition
The gambler’s fallacy is the mistaken belief that a streak of bad luck makes a person “due” for a streak
of good luck.
STREAKS
Another common misunderstanding that contributes to the gambler’s fallacy involves expectations
about streaks.
Suppose you toss a coin six times and see the outcome HHHHHH (all heads). Then you toss it six more
times and see the outcome HTTHTH.
Most people would say that the latter outcome is “natural” while the streak of all heads is surprising.
But, in fact, both outcomes are equally likely. The total number of possible outcomes for six coins is 2 × 2
× 2 × 2 × 2 × 2 = 64, and every individual outcome has the same probability of 1/64.
6.4 ideas of Risk and Life Expectancy
Here we will examine the ideas of probability can help us quantify rist, therby allowing us to make
informed decisions about such tradeloffs.
RISK AND TRAVEL
Often expressed in terms of an accidental rate or death rate.
VITAL STATISTICS
Data concerneing briths and deaths of citizens.
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LIFE EXPECTANCY
Often used to compare overall health at different times or in different countries.
Definition
Life expectancy is the number of years a person with a given age today can expect to live on
average.
6.5 Combining Probabilities – (Supplementary Section)
LEARNING GOAL
Distinguish between independent and dependent events and between overlapping and non-overlapping
events, and be able to calculate and and either/or probabilities.
AND PROBABILITIES
Suppose you toss two fair dice and want to know the probability that both will come up 4.
For each die, the probability of a 4 is 1/6. We find the probability that both dice show a 4 by multiplying
the individual probabilities:
P(double 4’s) = P(4) × P(4) =
In general, we call the probability of event A and event B occurring an and probability (or joint
probability).
And Probability for Independent Events
Two events are independent if the outcome of one event does not affect the probability of the
other event. Consider two independent events A and B with probabilities P(A) and P(B). The
probability that A and B occur together is
P(A and B) = P(A) × P(B)
This principle can be extended to any number of independent events. For example, the
probability of A, B, and a third independent event C is
P(A and B and C) = P(A) × P(B) × P(C)
And events can be independent or dependent
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And Probability for Independent Events
Two events are independent if the outcome of one event does not affect the probability of the other
event. Consider two independent events A and B with probabilities P(A) and P(B). The probability that
A and B occur together is
P(A and B) = P(A) × P(B)
This principle can be extended to any number of independent events. For example, the probability of
A, B, and a third independent event C is
P(A and B and C) = P(A) × P(B) × P(C)
And Probability for Dependent Events
Two events are dependent if the outcome of one event affects the probability of the other event. The
probability that dependent events A and B occur together is
P(A and B) = P(A) × P(B given A)
where P(B given A) means the probability of event B given the occurrence of event A.
This principle can be extended to any number of individual events. For example, the probability of
dependent events A, B, and C is
P(A and B and C) = P(A) × P(B given A) × P(C given A and B)
EITHER/OR PROBABILITIES
Either/Or Probability for Non-Overlapping Events
Two events are non-overlapping if they cannot occur at the same time. If A and B are nonoverlapping events, the probability that either A or B occurs is
P(A or B) = P(A) + P(B)
This principle can be extended to any number of non-overlapping events. For example, the
probability that either event A, event B, or event C occurs is
P(A or B or C) = P(A) + P(B) + P(C)
provided that A, B, and C are all non-overlapping events.Either/Or Probability for NonOverlapping Events
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Either/Or Probability for Overlapping Events
Two events A and B are overlapping if they can occur together. The probability that either A or B
occurs is
P(A or B) = P(A) + P(B) – P(A and B)
SUMMARY
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