2  Sample Spaces and Events

2.1 Sample Spaces

The sample space \(\Omega\) for a probability experiment is the set of all possible outcomes of the experiment.

Example: A single die is rolled and the number facing up is recorded. The sample space is \(\Omega = \{1,2,3,4,5,6\}\).

Example: A coin is tossed and the side facing up is recorded. The sample space is \(\Omega = \{H, T\}\).

Example: (Death of an insured) An insurance company is interested in the probability that an insured will die in the next year. The sample space is \(\Omega = \{death, survival\}\).

Not every sample spaces are so small or so simple.

Example: An insurance company has sold 100 individual life insurance policies. When an insured individual dies, the beneficiary named in the policy will file a claim for the amount of the policy. You wish to observe the number of claims filed in the next year. The sample space consists of all integers from 0 to 100, so \(\Omega = \{0,1,...,100\}\).

The sample space can also be infinite.

Example: A stock is purchased for \(\$100\). You wish to observe the price it can be sold for in one year. Since stock prices are quoted in dollars and fractions of dollars, the stock could have any non-negative rational number as its future value. The sample space consists of all non-negative rational numbers, \(\Omega = \{x : x > 0 \text{ and } x\in \mathbb Q\}\).

2.2 Events

An event is a subset of the sample space \(\Omega\).

For example, if a single die is rolled, the event “roll a number less than 5” consists of the outcomes in the set \(E = {1,2,3,4}\).

Example: You buy a stock for \(\$100\) and plan to sell it one year later. You are interested in the event \(E\) that you make a profit when the stock is sold. The event \(E\) is the subset of \(\Omega\) defined in the example above, and \(E = \{x : x > 100 \text{ and } x\in\mathbb Q\}\), the set of all possible future prices which are greater than the \(\$100\) you paid.