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Probability Key Definitions Help (page 2)

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By — McGraw-Hill Professional
Updated on Aug 26, 2011

Empirical Probability

In order to determine the likelihood that an event will have a certain outcome in real life, we must rely on the results of prior experiments. The probability of a particular outcome taking place, based on experience or observation, is called empirical probability.

Suppose we are told that a die is ''unweighted.'' How does the person who tells us this know that it is true? If we want to use this die in some application, such as when we need an object that can help us to generate a string of random numbers from the set {1, 2, 3, 4, 5, 6}, we can't take on faith the notion that the die is ''unweighted.'' We have to check it out. We can analyze the die in a lab and figure out where its center of gravity is; we measure how deep the indentations are where the dots on its faces are inked. We can scan the die electronically, X-ray it, and submerge it in (or float it on) water. But to be absolutely certain that the die is ''unweighted,'' we must toss it many thousands of times, and be sure that each face turns up, on the average, 1/6 of the time. We must conduct an experiment – gather empirical evidence – that supports the contention that the die is ''unweighted.'' Empirical probability is based on determinations of relative frequency, which was discussed in the last chapter.

As with mathematical probability, there are limits to the range an empirical probability figure can attain. If H is an outcome for a particular single event, and the empirical probability of H taking place as a result of that event is denoted pemp(H), then:

        0 ≤ pemp(H) ≤ 1
    0% ≤ pemp%(H) ≤ 100%

Real-World Empiricism

Empirical probability is used by scientists to make predictions. It is not good for looking at aspects of the past or present. If you try to calculate the empirical probability of the existence of extraterrestrial life in our galaxy, you can play around with formulas based on expert opinions, but once you state a numeric figure, you commit the PF. If you say the empirical probability that a hurricane of category 3 or stronger struck the U.S. mainland in 1992 equals x% (where x < 100) because at least one hurricane of that intensity hit the U.S. mainland in x of the years in the 20th century, historians will tell you that is rubbish, as will anyone who was in Homestead, Florida on August 24, 1992.

Imperfection is inevitable in the real world. We can't observe an infinite number of people and take into account every possible factor in a drug test. We cannot toss a die an infinite number of times. The best we can hope for is an empirical probability figure that gets closer and closer to the ''absolute truth'' as we conduct a better and better experiment. Nothing we can conclude about the future is a ''totally sure bet.''

Probability Key Definitions Practice Problems

Practice 1

Suppose a new cholesterol-lowering drug comes on the market. If the drug is to be approved by the government for public use, it must be shown effective, and it must also be shown not to have too many serious side effects. So it is tested. During the course of testing, 10,000 people, all of whom have been diagnosed with high cholesterol, are given this drug. Imagine that 7289 of the people experience a significant drop in cholesterol. Also suppose that 307 of these people experience adverse side effects. If you have high cholesterol and go on this drug, what is the empirical probability pemp(B) that you will derive benefit? What is the empirical probability pemp(A) that you will experience adverse side effects?

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