Final answer:
To calculate the probability of a sample space, divide the number of favorable outcomes by the total number of outcomes when all are equally likely. For conditional probabilities, restrict the sample space to the condition given and then calculate the probability within that restricted space.
Step-by-step explanation:
To find the probability of a sample space, you can use the fundamental principle that the probability of an event is the number of favorable outcomes divided by the total number of possible outcomes. This is assuming all outcomes are equally likely, which is often the case in theoretically perfect conditions like rolling a die or flipping a coin. For example, when rolling a fair, six-sided die, the sample space S is {1, 2, 3, 4, 5, 6}. The probability of rolling a 2 or a 3 can be found by recognizing that there are two favorable outcomes (2 and 3) out of the six possible outcomes. Therefore, the probability of rolling a 2 or a 3 is 2/6, which simplifies to 1/3.
For conditional probabilities, such as P(A|B), which is the probability of A given that B has occurred, the sample space is limited to only those outcomes that are also in event B. If event A is {2, 3} and event B is {2, 4, 6}, then P(A|B) will only consider outcomes that are in B. Since the only number that is in both A and B is 2, the conditional probability P(A|B) is 1/3, since there is 1 favorable outcome over 3 possibilities.