The Sample Space for the toss of a pair of dice.

Probability Spaces

By a probability space, we mean a sample space with a probability measure, that is, each subset within the sample space is assigned a probalility. For finite sample spaces with n points, we use Theorem 2 to assign probabilities. Since events are subsets of the sample space we want to investigate methods of assigning probability measures to events in terms of operations on sets. We begin with some examples.

Example

What is the probability of getting an 8 (as a sum) when you toss a pair of fair dice? In terms of sets, let A be the event of getting an 8 on the toss of a pair of dice. Notice A is composed of the five mutually exclusive outcomes O1 to O5. We write, A = O1 È O2 È ...È O5 and P(A) = P( O1) + P( O2) + ...+ P( O5) The probability of each outcome is 1/36, so P(A) = 5/36.

Example

Toss two dice. Find the probabilty that one of the dice shows a 1. Or by counting the sample points we check the result: 11/36.

Let A be the event that the black die comes up a one, and B the event that the white die comes up a one. Then the probability that we seek may be written: P(A È B) = P(A) + P(B) - P(A Ç B) we plug the values in and calculate P(A È B) = 1/6 + 1/6 - 1/36 = 11/36

Example

What is the probability of rolling a sum of 3 or a 6 with a pair of dice?
The event is illustrated by:

Again, it is easy to count and by Theorem 2 find the probability is 7/36.

Example

What is the probability that on rolling a white and black set of dice, the black die does not show a 1?

Easy enough? Sure as long as the sample spaces are small! When the sample spaces are not so easily managed, that is where our counting theorems and set theory really pay off.