Final answer:
The sample space when rolling a fair die is {1, 2, 3, 4, 5, 6}, with each outcome equally likely. Probabilities of events are calculated by dividing the number of favorable outcomes by the total number of outcomes in the sample space.
Step-by-step explanation:
When we roll a fair die, the sample space is the list of all possible outcomes. In this case, the sample space, S, is {1, 2, 3, 4, 5, 6}, representing the six faces of the die.
Since the die is fair, all outcomes are equally likely. To find the probability of a specific event, we count the number of outcomes that represent the event and divide by the total number of outcomes in the sample space. For example, if we define event A as rolling an odd number, A = {1, 3, 5}, there are three outcomes. Therefore, the probability of event A, P(A), is calculated as the number of outcomes in A divided by the total number of outcomes in the sample space: P(A) = 3/6 = 1/2.
To find the probability of rolling a number at least five, we look at event E with outcomes {5, 6}, which contains two elements, making the probability P(E) = 2/6 = 1/3. As another example, if we define event A = a prime number is rolled, we get A = {2, 3, 5}. The probability P(A) is then the number of outcomes in A divided by the total number of outcomes, so P(A) = 3/6 = 1/2.