Final answer:
A sample space in mathematical notation is denoted as a set listing all possible outcomes of an experiment. For flipping a coin, it's S = {H, T}, and for rolling a die, S = {1, 2, 3, 4, 5, 6}. Events are subsets of the sample space, like A = {2, 4, 6} for even numbers on a die, with the probability of an event calculated as the ratio of favorable to total outcomes.
Step-by-step explanation:
Defining a sample space in mathematical notation typically involves listing all possible outcomes of a random experiment within curly braces to form a set. This set represents the universal set of outcomes for that particular experiment.
For instance, consider the experiment of flipping a coin. The sample space, S, for a single coin flip with outcomes heads or tails can be denoted as S = {H, T}. If we extend this to the example of rolling a fair, six-sided die, the sample space would be S = {1, 2, 3, 4, 5, 6}.
When dealing with a compound experiment, such as flipping two coins, the sample space is constructed by listing all ordered pairs of the individual outcomes. For example, the sample space for flipping a fair dime and a fair nickel is S = {HH, HT, TH, TT}, where H and T represent "heads" and "tails" respectively.
Looking at an event within a sample space, it is typically represented by a capital letter such as A or B, and it contains subsets of the sample space. For example, if A represents getting an even number when rolling a dice, A would be expressed as A = {2, 4, 6}.
The probability of an event, P(A), is calculated based on how many outcomes in the event occur divided by the total number of outcomes in the sample space. For example, for event A when a die is rolled, P(A) = 3/6 since there are three favorable outcomes out of six possible outcomes.