660 CHAPTER 11 Probability In general, empirical probability is used when probabilities cannot be theoretically calculated. Empirical probabilities are used when there is enough data from past events to reasonably predict similar future events. For example, life insurance companies use data gathered over many years and empirical probability to determine the likelihood that an individual in a certain profession, with certain risk factors, lives to be age 75. Empirical probability is related to theoretical probability, which we will discuss later in this section, through the law of large numbers, which we will define shortly. It is reasonable to accept that if a “fair coin” is tossed many, many times, it will be heads approximately half of the time. Intuitively, we can guess that the probability that a fair coin will be heads is .1 2 Does that mean that if a coin is tossed twice, it will be heads exactly once? If a fair coin is tossed 10 times, will there necessarily be five heads? The answer is clearly no. What, then, does it mean when we state that the probability that a fair coin will be heads is ?1 2 To answer this question, let’s examine Table 11.1, which shows what may occur when a fair coin is tossed a given number of times. “The laws of probability, so true in general, so fallacious in particular.” Edward Gibbon, 1796 Table 11.1 Tossing a Coin Number of Tosses Expected Number of Heads Actual Number of Heads Observed Relative Frequency of Heads 10 5 4 = 4 10 0.4 100 50 45 = 45 100 0.45 1000 500 546 = 546 1000 0.546 10,000 5000 4852 = 4852 10,000 0.4852 100,000 50,000 49,770 = 49,770 100,000 0.49770 The far right column of Table 11.1, the relative frequency of heads, is a ratio of the number of heads observed to the total number of tosses of the coin. The relative frequency is the empirical probability, as defined earlier. Note that as the number of tosses increases, the relative frequency of heads gets closer and closer to , 1 2 or 0.5, which is what we expect. The nature of probability is summarized by the law of large numbers. Definition: Law of Large Numbers The law of large numbers states that probability statements apply in practice to a large number of trials, not to a single trial. It is the relative frequency over the long run that is accurately predictable, not individual events or precise totals. What does it mean to say that the probability of rolling a 2 with a single die is ?1 6 It means that over the long run, on the average, one of every six rolls will result in a 2. Next, we will study how empirical probability and the law of large numbers were used in the founding of the study of genetics. StatCrunch Applets Simulation Coin Flipping
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