Unit 7A

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Unit 7A

• Fundamentals of Probability

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BASIC TERMINOLOGY IN PROBABILITY

• Outcomes are the most basic possible results of
  observations or experiments.
• An event consists of one or more outcomes that
  share a property of interest.
• The probability of an event, expressed as
  P(event), is always a number between 0 and 1
  (inclusive).
• Many times upper case letters are used to
  represent events. For example, we could say that
  A is the event of getting heads when a coin is
  tossed. So, P(A) would be the probability of
  getting heads when a coin is tossed.

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PROBABILITY LIMITS

• The probability of an impossible event is 0.
• The probability of an even that is certain to
  occur is 1.
• Any event A has a probability between 0 and 1,
  inclusive.

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THE MULTIPLICATION PRINCIPLE
For a sequence of two events in which the first
event can occur M ways and the second event can
occur N ways, the events together can occur a
total of M N ways. This generalizes to more
than two events.
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EXAMPLES

• How many two letter words can be formed if the
  first letter is one of the vowels a, e, i, o, u
  and the second letter is a consonant?
• OVER FIFTY TYPES OF PIZZA! says the sign as you
  drive up. Inside you discover only the choices
  onions, peppers, mushrooms, sausage, anchovies,
  and meatballs. Did the advertisement lie?
• Janet has five different books that she wishes to
  arrange on her desk. How many different
  arrangements are possible?
• Suppose Janet only wants to arrange three of her
  five books on her desk. How many ways can she do
  that?

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THREE TYPES OF PROBABILITIES
There are three different types of probabilities.

• Theoretical Probabilities
• Empirical Probabilities
• Subjective Probabilities

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THEORETICAL PROBABILITY
A theoretical probability is based on a model in
which 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. That is,
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EXAMPLE
Find the probability of rolling a 7 when a pair
of fair dice are tossed.
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EMPIRICAL PROBABILITY
An empirical probability is based on observations
or experiments. It is the relative frequency of
the event of interest.
In baseball, batting averages are empirical
probabilities.
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COMPUTING AN EMPIRICAL PROBABILITY
Conduct (or observe) a procedure a large number
of times, and count the number of times that
event A actually occurs. Based on these actual
results P(A) is estimated as follows
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EXAMPLE
A fair die was tossed 563 times. The number 4
occurred 96 times. If you toss a fair die, what
do you estimate is the probability is for tossing
a 4?
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SUBJECTIVE PROBABILITY
A subjective probability is an estimate based on
experience or intuition.
EXAMPLE An economist was asked What is the
probability that the economy will fall into
recession next year? The economist said the
probability was about 15.
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PROBABILITY OF AN EVENT NOT OCCURRING
Suppose the probability of an event A is P(A).
Then the probability that the event A does not
occur is 1 - P(A).
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PROBABILITY DISTRIBUTION
A probability distribution represents the
probabilities of all possible events. It is
sometimes put in the form of a table in which one
column lists each event and the other column
lists each probability. The sum of all the
probabilities must be 1.
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MAKING A PROBABILITY DISTRIBUTION
To make a probability distribution
Step 1 List all possible outcomes. Step 2
Identify the outcomes that represent the same
event. Find the probability of each event using
the theoretical method. Step 3 Make a table in
which the first column lists each event and the
second column lists the probability of the event.
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EXAMPLES
1. Suppose you toss a coin three times. Let x be
the total number of heads. Make a table for the
probability distribution of x. 2. Suppose you
throw a pair of dice. Let x be the sum of the
numbers on the dice. Make a table for the
probability distribution of x.
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ODDS
The odds for (or odds in favor of) an event A are
The odds against (or odds on) an event A are
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ODDS IN GAMBLING
In gambling, the odds on usually expresses how
much you can gain with a win for each dollar you
bet.
EXAMPLE At a horse race the odds on the horse
Median are given as 5 to 2. If you bet 12 on
Median and he wins, how much will you gain?