6.1: Types 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 subjective in nature, in other words an educated guess. If someone asked you the probability that the Seattle Mariners would win their next baseball game, 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 they 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. 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. Let’s consider two other types of probabilities.

Another view of probability 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.

We will return to the subjective and experimental probabilities from time to time, but in this course we will mostly be concerned with theoretical probability, which is defined as follows:

Definition: Theoretical Probability

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 \(\dfrac{m}{n}\).
\[
\text { Probability of an event }=\dfrac{\text { number of ways the event can occur }}{\text { total number of possible outcomes }}
\]

Let's return to the coin flipping example. Instead of calculating the probability of the coin landing on heads by flipping a coin many times and observing the number of heads (which is an experimental probability) we could instead use what we know about coins. A fair coin has 2 equally likely possible outcomes: heads or tails. If we are interested in calculating the theoretical probability of a coin landing on heads, we could do so using the formula above: \(\dfrac{\text { number of ways a head can occur }}{\text { number of possible outcomes }}=\dfrac{1}{2}\).