4 Probability: The Study of Randomness

4.1 Randomness

Toss a coin, or choose an SRS. The result cannot be predicted with certainty in advance because the result will vary when you toss the coin or choose the sample again. But there is still a regular pattern in the results, a pattern that emerges clearly only after many repetitions. This remarkable fact is the basis for the idea of probability.

Reminder

simple random sample (SRS), p. 132

EXAMPLE 4.1 Coin Tossing

When you toss a coin, there are only two possible outcomes, heads or tails. Figure 4.1 shows the results of tossing a coin 5000 times twice. For each number of tosses from 1 to 5000, we have plotted the proportion of those tosses that gave a head. Trial A (solid line) begins tail, head, tail, tail. You can see that the proportion of heads for Trial A starts at 0 on the first toss, rises to 0.5 when the second toss gives a head, then falls to 0.33 and 0.25 as we get two more tails. Trial B, on the other hand, starts with five straight heads, so the proportion of heads is 1 until the sixth toss.

Figure 4.1: FIGURE 4.1 The proportion of tosses of a coin that give a head changes 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.

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The proportion of tosses that produce heads is quite variable at first. Trial A starts low and Trial B starts high. As we make more and more tosses, however, the proportion of heads for both trials gets close to 0.5 and stays there. If we made yet a third trial at tossing the coin 5000 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 appears as a horizontal line on the graph.

The Probability applet available on the text website animates Figure 4.1. It allows you to choose the probability of a head and simulate any number of tosses of a coin with that probability. Try it. As with Figure 4.1, you will find for your own trial that the proportion of heads gradually settles down close to the probability you chose. Equally important, you will find that the proportion in a small or moderate number of tosses can be far from the probability. Many people prematurely assess the probability of a phenomenon based only on short-term outcomes. Probability describes only what happens in the long run.

The language of probability

“Random” in statistics is not a synonym for “haphazard” but a description of a kind of order that emerges only in the long run. We often encounter the unpredictable side of randomness in our everyday experience, but we rarely see enough repetitions of the same random phenomenon to observe the long-term regularity that probability describes. You can see that regularity emerging in Figure 4.1. In the very long run, the proportion of tosses that give a head is 0.5. This is the intuitive idea of probability. Probability 0.5 means “occurs half the time in a very large number of trials.”

The idea of probability is empirical. That is, it is based on observation rather than theorizing. We might suspect that a coin has probability 0.5 of coming up heads just because the coin has two sides. Probability describes what happens in very many trials, and we must actually observe many trials to pin down a probability. In the case of tossing a coin, some diligent people have, in fact, made thousands of tosses.

EXAMPLE 4.2 Some Coin Tossers

The French naturalist Count Buffon (1707–1788) tossed a coin 4040 times. Result: 2048 heads, or proportion for heads.

Around 1900, the English statistician Karl Pearson heroically tossed a coin 24,000 times. Result: 12,012 heads, a proportion of 0.5005.

While imprisoned by the Germans during World War II, the South African mathematician John Kerrich tossed a coin 10,000 times. Result: 5067 heads, a proportion of 0.5067.

The coin-tossing experiments of these individuals did not just result in heads. They also observed the other possible outcome of tails. Pearson, for example, found the proportion of tails to be 0.4995. Their experiments revealed the long-term regularity across all the possible outcomes. In other words, they were able to pin down the distribution of outcomes.

distribution

Randomness and Probability

We call a phenomenon random if individual outcomes are uncertain but there is, nonetheless, a regular distribution of outcomes in a large number of repetitions.

The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions.

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Apply Your Knowledge

Question 4.1

4.1 Not just coins.

We introduced this chapter with the most recognizable experiment of chance, the coin toss. The coin has two random outcomes, heads and tails. But, this book is not about coin tossing per se. Provide two examples of business scenarios in which there are two distinct but uncertain outcomes.

Thinking about randomness and probability

Randomness is everywhere. In our personal lives, we observe randomness with varying outdoor temperatures, our blood pressure readings, our commuting times to school or work, and the scores of our favorite sports team. Businesses exist in a world of randomness in the forms of varying dimensions on manufactured parts, customers' waiting times, demand for products or services, prices of a company's stock, injuries in the workplace, and customers' abilities to pay off a loan.

Probability theory is the branch of mathematics that describes random behavior; its advanced study entails high-level mathematics. However, as we will discover, many of the key ideas are basic. Managers who assimilate these key ideas are better able to cope with the stark realities of randomness. They become better decision makers.

Of course, we never observe a probability exactly. We could always continue tossing the coin, for example. Mathematical probability is an idealization based on imagining what would happen in an indefinitely long series of trials. The best way to understand randomness is to observe random behavior—not only the long-run regularity but the unpredictable results of short runs. You can do this with physical devices such as coins and dice, but computer simulations of random behavior allow faster exploration. As you explore randomness, remember:

independence

You must have a long series of independent trials. That is, the outcome of one trial must not influence the outcome of any other. Imagine a crooked gambling house where the operator of a roulette wheel can stop it where she chooses—she can prevent the proportion of “red” from settling down to a fixed number. These trials are not independent.
The idea of probability is empirical. Computer simulations start with given probabilities and imitate random behavior, but we can estimate a real-world probability only by actually observing many trials.
Nonetheless, computer simulations are very useful because we need long runs of trials. In situations such as coin tossing, the proportion of an outcome often requires several hundred trials to settle down to the probability of that outcome. Exploration of probability with physical devices is typically too time consuming. Short runs give only rough estimates of a probability.