A General Discussion of ProbabilitySome “Probability Rules”

Some abstract math language too! (from various internet sources)

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Probability The Science of Random Behavior

• Random behavior is unpredictable for an individual object, but it has a regular and predictable pattern on the average for a huge number of objects.

• This is why we can use probability to gain useful results from random samples & randomized comparative experiments.

• Random Individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of repetitions.

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• Relative Frequency The proportion of occurrences of an outcome. It settles down to one value over the long run.

That one value is then defined to be the probability of that outcome.

• Relative Frequency Probabilities can be determined (checked) by observing a long series of independent trials (empirical data):– Do experiments with many samples– Do simulations, with computers, with random

number tables, etc.

Example: Flipping a Fair Coin

Probability Models• The sample space S of a random

phenomenon is the set of all possible outcomes.

• An event is an outcome or a set of outcomes (a subset of the sample space).

• A probability model is a mathematical description of long-run regularity consisting of a sample space S and a method of assigning probabilities to events.

The Probability Model for Two Fair DiceExample of Random Phenomenon: Roll a pair of fair dice.

The Sample Space is illustrated in the figure:

The probabilities of each individual of the 36 outcomes are found by inspection. Each clearly occurs with a probability:

p = (1/36) = 0.0278

Probability Rule #1: All Probabilities Must Be Numbers

Between 0 & 1.• A probability can be interpreted as the

proportion of times that a certain event can be expected to occur.

• So, if the probability of an event is more than 1, then it will occur more than 100% of the time!!

• This is clearly illogical & impossible!

Probability Rule #2: The sum of the probabilities of all

possible outcomes must be 1.• Because some outcome must occur on

every trial, the sum of the probabilities for all possible outcomes must be exactly one.

• If the sum of all of the probabilities is less than one or greater than one, then the resulting probability model will be incoherent & illogical.

Probability Rule #3:If two events have no outcomes in common, they

are said to be disjoint. The probability that one or the other of two disjoint events occurs is the

sum of their individual probabilities.• Example: Age distribution of women at first child birth:• Note: From discussion last time, the sample population is women

have had children! So, it excludes women with no children!

Under 20: 25%. 20-24: 33%. 25+: ?

Note: By Rule #3(or Rule #2)

This must be 42%

So, the probability that a woman in the sample population who is 24 or younger has had a first child is:

= 25% + 33% = 58%

Probability Rule #4:The probability that an event DOES NOT

OCCUR 1 minus the probability that the event DOES OCCUR.

• Example: As a jury member, you assess the probability that the defendant is guilty to be 0.8. So, you must also believe that the probability the defendant is not guilty is 0.2 in order to be consistent.

• Similarly, if the probability that a flight will be on time is 0.7, then the probability it will be late is 0.3.

Summary:Probability Rules in Mathematical Notation

Consider again the Random Phenomenon of rolling a pair of fair dice. As we’ve already seen, The Sample Space is illustrated in the figure:

The probabilities of each individual of the 36 outcomes are found by inspection. Each clearly occurs with a probability:

p = (1/36) = 0.0278

Calculate the probability P(5) of rolling a 5.P(5) P( ) + P( ) + P( ) + P( ) =(1/36) + (1/36) + (1/36) + (1/36) = (4/36) =

= 0.111

All possible outcomes of rolling a pair of fair dice are shown in the figure.