Transcript Chapter 14 - highlandstatistics

Chapter 14
Probability Time!
What is Probability?
A number.
 Specifically, it represents the long-term
likelihood of an event occuring.
 It is a model for what could happen, not
what will happen.
 It is often expressed as a percentage
when talking to people, but when doing
calculations it is usually left as a fraction
or a decimal.

How Do We Find Probability?
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There are a few different ways to find
probabilities.
A common one is in situations where there
is an equal chance for each event to happen.
◦ The probability of drawing a specific card from a
deck is 1/52 (assuming the jokers were removed).
◦ The probability of getting a specific number when
you roll a random number cube is 1/6.
◦ The probability of heads on a fair coin is 1/2.
How Do We Find Probability?

Another way to find probability is to
create a fraction of the number of times a
specific outcome happens divided by all of
the outcomes.
◦ For example, if I have a bag with 20 marbles
and 5 of them are red, the probability of
drawing a red marble if one is chosen at
random is 5/20, or 1/4.
◦ The probability of rolling a prime number on
a random number cube is 3/6, or 1/2.
How Do We Find Probability?
Yet another way to find probabilities is if
we are just told them.
 As in, if we are told that P(A) = .3 in the
problem.
 This method of finding probabilities is
probably the easiest.
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How Do We Find Probability?
A similar way to find probability is to take
a known probability and use the rules of
probability to calculate an unknown
probability.
 For example, if we wanted to know Ac
and we know that P(A) = .3, then P(Ac) =
1 – P(A) means that P(Ac) = 1 – .3 = .7.
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How Do We Find Probability?
One more way is that we guess.
 Mind you, this does not mean guess
casually or haphazardly.
 This means guessing by bringing our life
experiences and finely honed intuition to
bear.
 So yeah, this is one more thing in
Statistics we use our intuition for.
 This is called a personal probability.

How Do We Find Probability?

The final way we will discuss is to take a
sample and estimate.
◦ In other words if we want to guess at what
our probability of making a free throw in
basketball is, we could always just grab a
basketball, attempt ten free throws in a row,
and then ballpark our true probability based
on those ten throws.

This is called an experimental probability.
Finding Probability Summary
Equally likely outcomes
 Determine the fraction
 They tell us
 Using rules to calculate
 Personal probabilities
 Experimental probabilities

Probability In Action
Once we have determined a probability,
we need to keep in mind what it stands
for.
 It stands for the likelihood that a
particular thing happens.
 To avoid overusing the word thing, we are
going to use some specific language.
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Basic Terminology
Random Phenomena – Random things for
which we know all possible outcomes, but
we do not know which one will actually
happen.
 Phenomena is the plural of phenomenon.
 This definition also covers things which
have unpredictable outcomes as long as
one of the outcomes we consider is some
kind of a “miscellaneous” or “other”
category.

Basic Terminology

Trial – This is what we call one instance of
the random phenomenon.
◦
◦
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One card is drawn from a deck.
A random number cube is rolled once.
A coin is flipped once.
A marble is pulled out of the bag.
Multiple trials can be certainly be done, but
the term trial refers to just a single instance.
◦ A coin flipped three times in a row is three trials.
Basic Terminology

Outcome – This is the end result of a trial.
◦ This is the number showing on the random
number cube.
◦ This is the side showing on the coin.
◦ This is the particular card drawn from the deck.
◦ This is the color of the marble pulled out of the
bag.

The exact outcome that will occur is
unknown before the trial.
Basic Terminology
Event – This is an outcome or collection
of outcomes that we are interested in.
 Events are typically labeled with capital
letters, starting with A.
 If you are typing, they will usually be
bolded as well.
 Mr. Sanford rarely bothers with this, and
will not expect you to bother with it
either.
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Basic Terminology
Complement – The complement of an
event is the set of all outcomes that are
not part of the event.
 Since every outcome is either part of an
event or not part of an event, when you
put an event together with its
complement, you get all possible
outcomes.
 This is why we use the word
complement.
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Basic Terminology
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Disjoint – These are two or more events
that cannot happen at the same time.
◦ When you roll a die, you will not get a 3 and a 5
on the same roll.
◦ When you flip a coin you will not get heads and
tails.
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An event and its complement are always
disjoint.
Another way to say disjoint is mutually
exclusive.
◦ This gets said a lot more often than you might
realize, so you ought to learn it as well.
Basic Terminology
Dependent – Two events are dependent if
knowing that one happened changes the
probability of the other.
 The probability of getting a ticket for
speeding and how much you are speeding
by are dependent.

◦ If you are not speeding at all, the probability is
very low.
◦ If you are going 15 mph over the speed limit,
the probability is higher.
Basic Terminology
Independent – Two events are
independent if knowing one variable
turned out does not affect the other.
 This is a somewhat informal definition.
 We will define independence a little
differently later.
 Would knowing someone’s favorite color
give you any insight into knowing
someone’s favorite number?

Probability Rules
Rule: 0 ≤ P(x) ≤ 1 for each outcome or
event.
 Rule: The sum of all possible outcomes
must equal 1.

◦ This does not have to mean that the sum of
all events equals 1. It is specifically outcomes.

Rule: P(Ac) = 1 – P(A)
Probability Rules
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Rule: When events are disjoint, we can
add their probabilities to calculate the
probability of the union of those events.
◦ P(A U B) = P(A) + P(B)
disjoint)
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(if A and B are
Rule: When events are independent, we
can multiply their probabilities to
calculate the probability of the
intersection of those events.
◦ P(A ∩ B) = P(A)•P(B)
independent)
(if A and B are
Last Thing
To determine if a probability assignment is
legitimate we need to verify two things.
 We need to verify that each outcome has
a probability between 0 and 1, inclusive.
 We also need to verify that all of the
outcomes add to a total probability of 1.
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Assignments
Chapter 14 homework will be released in
three waves.
 First wave: one problem from 1-5, and
problems 10 and 24.
 Due date is forthcoming.
 There will be a quiz over chapters 14 and
15 eventually.
 Bulletpoints will be forthcoming.
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