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genetic makeups

Because of their comparative simplicity, experiments with finite sample spaces are discussed first. In the early development of probability theory, mathematicians considered only those experiments for which it seemed reasonable, based on considerations of symmetry, to suppose that all outcomes of the experiment were “equally likely.” Then in a large number of trials all outcomes should occur with approximately the same frequency. The probability of an event is defined to be the ratio of the number of cases favourable to the event—i.e., the number of outcomes in the subset of the sample space defining the event—to the total number of cases. Thus, the 36 possible outcomes in the throw of two dice are assumed equally likely, and the probability of obtaining “six” is the number of favourable cases, 5, divided by 36, or 5/36.

Now suppose that a coin is tossed n times, and consider the probability of the event “heads does not occur” in the n tosses. An outcome of the experiment is an n-tuple, the kth entry of which identifies the result of the kth toss. Since there are two possible outcomes for each toss, the number of elements in the sample space is 2n. Of these, only one outcome corresponds to having no heads, so the required probability is 1/2n.

It is only slightly more difficult to determine the probability of “at most one head.” In addition to the single case in which no head occurs, there are ncases in which exactly one head occurs, because it can occur on the first, second,…, or nth toss. Hence, there are n + 1 cases favourable to obtaining at most one head, and the desired probability is (n + 1)/2n.