Lesson 2: Introduction to Probability

Relative Frequency Assignment

A second, and most often used method, of assigning probabilities of occurrence to experimental outcomes is the relative frequency method. In this method, you use the results of an experiment to determine what probabilities to assign.

Impossible Events

An event that cannot possibly occur has a probability of zero associated with it. P(A) = 0

Certain Events

An event that is certain to occur has a probability of 1.0 associated with it. P(A) = 1

P(A) means the probability of an event A

.It does not mean P times A.

An auto dealer collected weekly sales data over a period of 50 weeks as follows. Sales: 0 1 2 3 4 5; Frequency 5 11 14 11 6 3 (Example: there were 5 weeks with 0 sales, 11 weeks with 1 sale, etc.). What are the corresponding probability assignments?

0.10, 0 .22, 0.28, 0.22, 0.12, 0.06 These are the correct probability assignments based on the given frequencies. For example, we get 5/50 = 0.10, 11/50 = 0.22, etc.

In order to compare two or more probabilities, you need to have a numerical measure of each probability.

For example, it is not enough to say that event A1 is more likely to occur than event A2. You have a lot more information if you know that event A1 is twice as likely to occur as event A2.

The first point to keep in mind is that probabilities are inherently estimates. So you are not looking for precision.

For example, it would be unrealistic to state that your estimate of P(A) = 0.4378 (regardless of what the event A consists of). There is no way that you can assign a probability with that degree of accuracy except for an unusual situation in which you were able to set up a repeatable experiment with a low margin of error.

I'm not clear about the difference between independent events and mutually exclusive events. If one mutually exclusive event is known to occur, the probability of the other occurring is zero.

For example, let A be the event that you complete this course, and B be the event that you don't complete this course.These are two mutually exclusive events, in that they cannot both be true. Therefore if P(A) is not = 0, then P(B) must = 0. This means that these two events cannot be independent.

So what are you looking for in making probability assignments? I would say you are looking for a reasonable estimate, which you can determine by asking questions such as "is P(A) closer to 0.2 or to 0.8?"

Frequently, the nature of the situation will make it clear to you what the answer is to that question.

Which of the following are the possible outcomes of an experiment where two coins are tossed once simultaneously. H stands for heads, and T stands for tails.

HH, HT, TH, TT These outcomes are the four possible outcomes if two coins are tossed once simultaneously.

How do I assign probabilities to business situations?

I discussed two methods of assigning probabilities in Chapter 2: the classical probability assignment and the relative frequency assignment. I believe you meant to ask specifically how you can develop probability estimates using the relative frequency assignment.

Possible Occurrences

If there are m ways in which an event can occur out of a total of n ways, then the probability that m occurs is given by m/n.

Example of experiment

If you run a newspaper advertisement, you are conducting an experiment to see how effective the ad is in achieving your purpose, such as to bring new customers into the store.

Classical Probability Assignment

In the classical method you assign equal probabilities to each possible event. For example, if you flip a coin, there is an equal probability of heads or tails. This, of course, assumes that the coin is 'honest', that is, not unbalanced in some way.

If A and B are two independent events, then the probability that both events occur is given by the product of the probability of each event; that is P(A and B) = P(A) x P(B)

Read this equation as: the probability that two independent events A and B occur is the product of the probabilities if A occurs and B occurs.

Bounded by Zero and One

So any probability value must be between zero and one, including the possibility of zero and one. 0 < or=P(A) < or = 1.0

Sum of Probabilities

The sum of all the probabilities of every disjoint event must equal 1.P(E1) + P(E2) + P(E3) + P(E4) + P(E5) = 1Or 0.2 + 0.2 + 0.2 + 0.2 + 0.2 = 1 Refer to all the possible outcomes of an experiment as the universe of possibilities

Probability Assignments

To conduct a probability experiment, you must first assign probabilities of occurrence to experimental outcomes. You may use various methods to make an assignment.

independent events

Two or more events in which the outcome of one event does not affect the outcome of the other event(s).

Let's now consider independent events. Let A be the event that you complete this course, and B be the event that you change jobs.

You must decide if there is any logical connection between these two events. If there is no such connection, then these two events are independent. That is, your completion of this course and your change of jobs are in no way related to one another.

Your answer will be based on experience, common sense, and possibly even an appropriate test.

You will find that over time, you will make more effective decisions by use of probability assignments than by simply shrugging your shoulders and saying, "I haven't a clue."

Probability is

a numerical measure of the likelihood that an event will occur.

A trial is

an experiment conducted one time. If you repeat the experiment, then you conduct a second trial, and so on to whatever number of trials of the experiment that you conduct.

An experiment is

any process of trial and observation in which the outcome is determined by chance. Experiments play a key role in probability.

mutually exclusive events

events that cannot happen at the same time. In other words, there are no elements in common between the events.P(A) = 1 - P()

Example of event

in response to a newspaper ad you may get 50 new customers. This is an event, which is an outcome of your experiment in which you ran the ad. Another event may be that you get 18 new customers. Unfortunately, in the difficult world of business, you may get no new customers, which is also a possible event.

probability relationship is that probabilities are

non-negative.P(A) > or = 0

An event is

the occurrence of either a prescribed outcome or any one of a number of possible outcomes of an experiment