# Probability Concepts and Discussion

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1. Define probability and explain its three perspectives. Provide an example of each.

2. Explain the concept of mutually exclusive events. How do you compute the probability P(A or B) when A and B are, and are not, mutually exclusive.

3. What is the standard error of the mean? How does it relate to the standard deviation of the population from which a sample is taken?

4. Describe the difference between subjective and probabilistic sampling methods. What are the advantages and disadvantages and disadvantages of each?

5. How does the t-distribution differ from the standard normal distribution?

6. Discuss how confidence intervals can help in making decisions. Provide some examples.

#### Solution Preview

1. Define probability and explain its three perspectives. Provide an example of each.

The three "perspectives" of probability (and that's a weird way to state it) are: (1) the Classical Method; (2) the Relative Frequency Method; and, (3) the Subjective Method.

In the Classical approach you assume that all outcomes are equally likely, and so if there are 8 equally likely outcomes the probability of any one of them occurring is 1/8. An example is rolling a fair 6-sided die. The probability that you roll a 4 is 1/6; the probability that you roll a 1 is 1/6, and so on.

In the Relative Frequency approach you don't really make any assumptions at all. Instead, you perform the experiment in question a large number of times (say 200 times) and count how many times the event of interest actually occurs out of this 200 times (say that it occurs 50 times out of 200 times that you repeated the experiment). Then, you may estimate the probability that the event of interest occurs by dividing the number of times that it DID occur by the number of times that the experiment was performed ... in our example 50/200 = 0.25. A specific example of this approach would be trying to assign a probability to taking a basketball free-throw. When you attempt a free-throw you either make it or you don't, but to say that there's a 50% probability that you make it is probably not correct for most of us (and that IS what a Classical approach would say). Taking a Relative Frequency approach you could attempt 500 free-throws from the foul line and count how many you actually made ... let's say that you made 400 out of the 500 you attempted. Then the Relative Frequency approach estimate of the probability that you make a basketball free-throw would be 400/500 = 0.80.

In the Subjective approach you pretty much make your probability determination based on your gut instinct. In this situation you don't have a whole lot of other information to go on except perhaps for previous experience, gut instinct, a hunch ... an example of this approach might be your assessment of the probability that all 3 job interviews that you went on will lead to a job offer. It seems unreasonable to assume that the probability that each interview leads to a job offer is the same (which is what the Classical approach would assume) and you're not going to be able to go to each job interview a large number of times (which is what the Relative Frequency approach would require). Pretty much all you've got is your feeling of how each one went in order to come up with the probability that all 3 lead to a job offer. This is a totally subjective determination.

2. Explain the concept of ...

#### Solution Summary

The expert defines probability and explains three perspectives. The concepts of mutually exclusive events are explained.