Chapter 2
Probability, Counting & Binomial
2 Probability
Probability is a numerical measure of the likelihood that an event will occur.
Probability is the study of chance. e.g.
What is the chance that I will pass Stats?
What is the chance that it will rain tomorrow?
What is the chance of an earthquake in LA this year?
What is the chance of getting 0 on the roulette wheel?
We want to attach a number to each of the above examples.
Probability values are always assigned on a scale from 0 to 1.
A probability near zero indicates an event is quite unlikely to occur.
A probability near one indicates an event is almost certain to occur.
Hopefully, the chance of passing this class will be 100%.
Terminology
Random Experiment
An action or process that leads to one of several possible outcomes.
E.g. rolling a pair of dice.
Outcome/Sample Point
A particular result of an experiment.
E.g. rolling pair of sixes with two dice.
Sample Space
All the possible outcomes of an experiment.
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Chapter 2
Probability, Counting & Binomial
Event
Outcomes that are of interest for an experiment. They are a subset of the sample space. A
collection or set of one or more simple events in a sample space.
The probability of an event occurring is equal to the sum of the probabilities of the
outcomes giving rise to that event.
If we identify all the outcomes of an experiment and assign a probability to each, we can
calculate the probability for that event.
Complement of an Event
The complement of event A is defined to be the event consisting of all outcomes that are
not in A. The complement of A is denoted by , and is referred to as A prime.
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Chapter 2
Probability, Counting & Binomial
Classical Probability
This occurs when we know all the possible outcomes to an experiment and the number of
outcomes for a particular event. Here the probabilities are assigned on the assumption of
equally likely outcomes.
The probability of a particular event, A, occurring is denoted P[A] .
P[A] = Number of possible outcomes in which event A occurs
Total number of outcomes from the sample space
So if an experiment has n possible outcomes, the classical method would assign a
probability of
to each outcome.
Example
Consider the experiment of rolling 1 die.
The sample space will look like this
(1)
(2)
(3)
(4)
(5)
(6)
This experiment has a total of 6 outcomes. The total outcomes form the sample space.
P[rolling of rolling any of the possible outcomes] =
1
6
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Chapter 2
Probability, Counting & Binomial
Example
Consider the experiment of rolling 2 dice.
Consider the event of rolling a total of 3,4,5 or 6 with 2 dice.
The sample space will look like this
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
(1,6)
(2,1)
(2,2)
(2,3)
(2,4)
(2,5)
(2,6)
(3,1)
(3,2)
(3,3)
(3,4)
(3,5)
(3,6)
(4,1)
(4,2)
(4,3)
(4,4)
(4,5)
(4,6)
(5,1)
(5,2)
(5,3)
(5,4)
(5,5)
(5,6)
(6,1)
(6,2)
(6,3)
(6,4)
(6,5)
(6,6)
This experiment has a total of 36 outcomes. The total outcomes form the sample space.
We have 14 outcomes, underlined, corresponding to our event of rolling 3, 4, 5 or 6 with
2 dice.
Applying the formula gives
P[rolling 3,4,5 or 6 with 2 dice] =
14
= 0.39
36
The 0.39 could be expressed as 39.0%
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Chapter 2
Probability, Counting & Binomial
Empirical Probability/Relative Frequency Method
Empirical comes from the Greek word meaning “experience”. Probabilities are assigned
based on experimentation or historical data.
This method may use data built over past experience e.g. the motor claims for a particular
company. We do not totally understand the underlying process of who will crash their car
or have it stolen.
We just take the actual past data, our experience, and work out the percentage of people
that have crashed their cars or had them stolen.
To work out the empirical probability we observe the number of occurrences of an event
through an experiment and calculate the probability from a relative frequency
distribution:
P[A] = Frequency in which event A occurs
Total number of observations
Subjective Probability
Used when other two methods are not available. We base our numbers on guesstimates.
This may be based on intuition or gut feel.
E.g. a new type of insurance for domestic pets that has never been sold before. We have
no experience of past claims so we need to subjectively arrive at a premium.
Requirements/Properties of Probability
If P[ A ] = 1 (100%) then event A will occur with certainty.
If P[ A] = 0 (0%) then event A will not occur.
Probability of event A must be between 0 and 1 (0% to 100%).
The sum of all the probabilities for the events in the sample space must equal 1 (100%).
The complement to an event A, are all events in the sample space, which do not include
event A. The complement of A is referred to as A ′ .
P[ A] + P[ A′] = 1
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Chapter 2
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The rule is useful as sometimes it is easier to calculate the probability of an event
happening by determining the probability of it not happening and then subtracting the
result from 1.
Example
An automatic machine fills plastic bags with a mixture of beans, broccoli and other
vegetables. Most of the bags contain the correct weight, but because of the variation in
the size of beans and other vegetables, a package may be underweight, overweight or
have the correct weight.
A check of 4,000 packages resulted in the following results:
Probability A, of being underweight is 0.025 or 2.5%
Probability B, of having satisfactory weight is 0.90 or 90%
Probability C, of being overweight is 0.075 or 7.5%
What is the probability of choosing a satisfactory weight bag using the complement rule?
The probability the bag is unsatisfactory equals the probability the bag is overweight or
underweight.
That is P ( AorC ) = P ( A) + P (C ) = 0 .025 + 0 .075 = 0 .100
The bag is satisfactory if it is not underweight or overweight, so
P ( B ) = 1 − [ P ( A) + P (C )] = 1 − [0 .025 + 0 .075 ] = 0.900
This result can be seen from the Venn diagram below.
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