Download Chapter 14: Probability

Chapter 5
The Idea of Probability
AP Statistics
The Point of this Chapter
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When we do an experiment or observational
study, we would like to know if our results are
statistically significant.
That is, we want to know if the results we
obtained were likely to occur simply by chance.
To determine if our results are statistically
significant, we need calculate the probability,
which is what we will study in the next chapter.
Definition of Probability
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The study of probability is the systematic study
of uncertainty.
Probability is the long run relative frequency of
an occurrence.
A chance experiment is any activity or situation
in which there is uncertainty about which of two
or more possible outcomes will result.
 ex:
flipping a coin, rolling a die, choosing a card,
asking a survey question, giving a treatment
Understanding Probability
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An event is any collection of outcomes from the
sample space of a chance experiment.
 Note:
We usually denote events with capital letters:
A, B, C, ...or a capital letter with subscripts:
 ex: A = choosing a senior male = {senior male from
2nd period, senior male from 3rd period or senior
male from 4th period}
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A simple event is an event consisting of exactly
one outcome.
 ex:
B = choosing a junior female from 3rd period =
{junior female from 3rd period}
 ex: find an event C such that C has three outcomes.
List the outcomes.
Understanding Probability
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The probability that an event will occur is
expressed as a number between 0 and 1.
The following are examples of the notation we
use for probability:
 P(A)
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represents the probability that the event A will occur
 P(A)
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represents the probability that event A will occurs is 1 or
that the probability that event A occurs is CERTAIN
 P(A)
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=1
=0
represents the probability that event A will occurs is 0 or
that the probability that event A occurs is IMPOSSIBLE
Understanding Probability
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The collection of all possible outcomes of a
chance experiment is the sample space for the
experiment.
 ex:
flipping a coin: heads, tails
 ex: rolling a die: 1, 2, ... 6
 ex: choosing a card: ace of hearts, 2 of hearts, .....king
of spades (52 total)
Understanding Probability
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In some chance experiments, more than one
piece of data is collected.
 ex:
a randomly selected stats student will be asked
for his or her gender, class, and period.
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Thus, one possible outcome would be a Senior Male in 3rd
period.
We can use a tree diagram, with each set of branches
corresponding to one variable and each “end” representing
one outcome.
How many possible outcomes are there?
To identify all possible outcomes, we may wish to use the
Fundamental Counting Principle:
 If there are n ways that Event A can occur and m ways
that Event B can occur, then there are m*n ways that
both can occur
More Definitions
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For any random phenomenon, each attempt, or
trial, generates an outcome. Combinations of
outcomes are called events.
 Once
again, a phenomenon consists of trials. Each
trial has an outcome. Outcomes combine to make
events.
 Things are simplified as long as each trial is
independent – generally speaking, this means that
one trial doesn’t affect the outcome of another.
Always?
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Will the long run average always give us a regular
distribution?
Yes. In fact, this is one of the most important
concepts to probability, a principle called the Law of
Large Numbers (LLN).
The Law of Large Numbers says that as the number
of trials increases, the relative frequency of an event
will approach the true probability of the event
The LLN guarantees that the long-run relative
frequency of repeated independent events settle
down to the true probability as the number of trials
increases.
The LLN and the Gambler’s Fallacy
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People often misunderstand or misuse the Law
of Large Numbers because the idea of the long
run is so abstract.
Many people often think that after seeing a
series of similar events (such as six heads in a
row when flipping a coin) suggests that an
alternate event is “due” (after six heads, there is
a greater chance of getting tails since it is
“due”) – however this is an error called the
Gambler’s Fallacy.
The LLN and the Gambler’s Fallacy
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The problem is that the LLN doesn’t apply in
short-run behavior. The probability only evens
out in the long-run! In fact, it must be infinitely
long before we are sure to get the true
probabilities.
 Just
because you flip a coin and get heads six times in
a row, doesn’t mean that a tails is due. The coin
doesn’t hold a memory and make things come out
right!
 In fact, in a very long series, you would see many
streaks of heads, but afterwards, there does not need
to be a streak of tails.
A Strategy for Winning the Lottery
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A common proposal for beating the lottery is to
note which numbers have come up lately,
eliminate those from consideration, and bet on
numbers that have not come up for a long time.
Proponents of this method argue that in the long
run, every number should be selected equally
often, so those that haven’t come up are due.
 Is
this a good strategy…Why or why not?
A Strategy for Winning Keno
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A group of graduate Statistics students decided
to take a trip to Reno. They very discretely
wrote down the outcomes of the game for a
couple of days (several hundred runs of the
game), then they drove home and tested
whether or not the numbers were equally likely.
It turned out that some numbers showed up
more often than others. Rather than betting on
the numbers that were “due,” they conjectured
that according to the LLN, certain numbers were
more likely than others and betted on those
numbers. They took $5000 and put their theory
to the test.
A Strategy for Winning Keno
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Did the Statistics students have a good strategy?
Why or why not?
So what happened?
 The
students ended up pocketing over $50,000. They
were escorted off the premises, and asked to never
show their faces in that casino again.
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Why was the first example an instance of the
gambler’s fallacy and the second example a
proper use of the LLN?
Probability
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Finding probabilities are easy as long as each
event is equally likely. Be careful when they are
not. Consider the following interview:
 Interviewer:
What do you think your chances are of
winning the lottery?
 Lottery ticket purchaser: Oh, about 50-50.
 Interviewer: What?! How did you get that?
 Lottery ticket purchaser: Well, I figure that either I
win or I don’t!
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The moral of the story is that outcomes are not always
equally likely!