Example 1

EXAMPLE 1 Heads Up When Tossing a Coin: Long-Run Frequency Interpretation of Probability

When you toss a coin, there are only two possible outcomes: heads or tails. Figure 8.3 shows the results of tossing a coin 5000 times twice. Let’s focus on Trial A, the red graph. For Trial A, the first four tosses result in tail, head, tail, tail. After four tosses, the proportion of heads is . Notice that corresponding to the first 100 tosses, there is quite a bit of fluctuation in the proportions. Now, compare the amount of fluctuation about the horizontal black line for relatively few tosses, say between 1 and 100 tosses, with the amount of fluctuation corresponding to relatively many tosses, say between 2000 and 5000 tosses. Comparatively, there is very little fluctuation in the latter interval.

Next, compare the proportions of heads for Trial A (red graph) in Figure 8.3 with those plotted for Trial B (blue graph). Trial B starts with five straight heads, so the proportion of heads is 1 until the sixth toss. Notice that the proportion of tosses that produces heads for both Trials A and B is quite variable at first. Trial A starts low and Trial B starts high. As we make more and more tosses, however, the proportions of heads for both trials get close to 0.5 and stay there. If we made yet a third trial at tossing the coin a great many times, the proportion of heads would again settle down to 0.5 in the long run. We say that 0.5 is the probability of a head. the probability 0.5 is marked by the horizontal line on the graph.

344

Figure 8.3: Figure 8.3 The proportion of tosses of a coin that gives heads varies as we make more tosses. Eventually, however, the proportion approaches 0.5, the probability of a head. This figure shows the results of two trials of 5000 tosses each. (the horizontal scale is transformed using logarithms to show both short- and long-term behavior.)