4.1: Introduction to Probability
- Page ID
- 74315
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- Describe theoretical, empirical, and subjective probability
- Distinguish among the three uses of probability
The probability of a specified event is the chance or likelihood that it will occur. There are several ways of viewing probability.
One would be experimental in nature, where we repeatedly conduct an experiment. Suppose we flipped a coin over and over and over again and it came up heads about half of the time; we would expect that in the future whenever we flipped the coin it would turn up heads about half of the time. When a weather reporter says “there is a 10% chance of rain tomorrow,” she is basing that on prior evidence; that out of all days with similar weather patterns, it has rained on 1 out of 10 of those days.
Example 1
Conduct an experiment to determine the probability of the spinner landing on 1?
Solution
Using the experimental method, suppose out of 10,000 spins, 1,711 of those landed on 1. This is normally done on a computer application that can randomly generate outcomes of each spin to avoid someone having to spin the spinner 10,000 times.
To calculate this probability, divide the number of times 1 has occurred which is 1,711 by the number of times the spinner was spun which was 10,000.
The result is 0.1711 or approximately 17% of the time. So 17 times out of 100 spins one would expect the spinner to land on the number 1.
Another view would be subjective in nature, in other words an educated guess or opinion. But this is just a guess, with no way to verify its accuracy, and depending upon how educated the educated guesser is, a subjective probability may not be worth very much.
Example 2
Determine the probability that the Seattle Mariners would win their next baseball game.
Solution
It would be impossible to conduct an experiment where the same two teams played each other repeatedly, each time with the same starting lineup and starting pitchers, each starting at the same time of day on the same field under the precisely the same conditions.
Since there are so many variables to take into account, someone familiar with baseball and with the two teams involved might make an educated guess that there is a 75% chance the Mariners will win the game; that is, if the same two teams were to play each other repeatedly under identical conditions, the Mariners would win about three out of every four games.
Definition: Probabilities
Empirical Probability uses the results of an experiment to predict the percent chance an event could occur.
Subjective Probability uses intuition or guesswork to predict the percent chance an event could occur.
Theoretical Probability uses the number of possible desired outcomes of an event compared to the number of all possible outcomes of an event to predict the percent chance an event could occur.
Theoretical Probability is defined mathematically as follows:
Suppose there is a situation with \(n\) equally likely possible outcomes and that m of those \(n\) outcomes correspond to a particular event; then the probability of that event is defined as \(\frac{m}{n}\).
We will return to the empirical and subjective probabilities from time to time, but in this course we will mostly be concerned with theoretical probability.