Overview:
Probability is the measure of the likelihood of an event. The basic mathematics of probability theory started with games of chance, but it can be applied to many situations, from weather forecasting to politics. Probabilities range from 0 (no likelihood) to 1 (certainty), and are expressed as rational numbers.
What Is the Sample Space?
The sample space is the set of all possible outcomes of an event. For a coin toss, the coin will either result in heads or tails. For the roll of one die, the sample space is all the values on the faces of the die, or a set of {1, 2, 3, 4, 5, 6}. For the roll of a pair of dice, the sums will be in a set from 2 (both dice give you a 1), the smallest sum possible, to 12 (both dice give you 6), the largest sum possible.
What Is A Fair Experiment?
In a fair experiment, all possible outcomes are equally likely. The probability of any outcome is related to the total number of outcomes by a ratio of the number of outcomes in that event to the number of all possible outcomes of the event (the sample space). Therefore, the probability that a coin will be heads is 1/2. The probability that if one die is rolled, the number on top will be a 3 is 1/6.
What If Events Are Not Equally Likely?
Sometimes, possible outcomes can be combined in such a way so that not all outcomes are equally likely. Suppose two fair coins are tossed: there are 4 possibilities in the sample space {HH, HT, TH, TT}. The probability of each event when order is important equals 1/4 for each possibility. However, if the question is merely “How many heads come up when two coins are tossed?”, there are only 3 possibilities in the sample space, 0 heads, 1 head, or 2 heads. The event 0 heads is defined as {TT}, and the event 2 heads is defined as {HH}. However, there are two possibilities for 1 head, either {HT} or {TH}, so the outcomes are not equally likely.
What Are Mutually Exclusive Events?
If events are mutually exclusive, it means that neither sample space A or sample space B contain common elements. Therefore, the probabilities can be added to form the probability of one event or the other occurring. The sample space for tossing two fair dice and getting a sum of 7 consists of {(6,1), (1, 6), (2, 5), (5,2), (3,4), and (4,3)}. The sample space for tossing two fair dice and getting a sum of 11 consists of {(6,5), (5,6)}. The sample space for getting either a sum of 7 or a sum of 11 is the union of both sets.
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