SPCMichael J. Cleary, Ph.D.
 Heads, You WinFlipping a coin and predicting stability

Hartford Simsack, quality manager for Greer Grate & Gate, finally has things in his department under control. That is, operators and others have stopped asking complex statistical questions he can't answer, and this pleases him. No longer must he be on the defensive, stalling until he can check a statistics text or ask his mentor, Dr. Stan Deviation.

Now, to establish his reputation as an expert in statistical process control, he simply drops impressive technical terms (picked up from the index of his statistics book) that may confuse others but make him feel superior. His co-workers hear him mumbling about "checking the Cpk and the Ppk" and "noting the kurtosis," but they know better than to ask him to elucidate.

Bjorn Luzer, the newest member of Simsack's team, doesn't yet realize the limits of his boss's statistical knowledge. He shows Simsack an X-bar and R chart with seven X-bars below an X-double bar, pointing out that seven points below the process average represents a sign that the process is out of control, or indicates a special cause. He's wondering now why seven is significant in this way, rather than eight or six. "Is this simply part of our heritage of medieval numerology?" he asks, noting that seven is a culturally significant number and wondering if it also has statistical significance.

Simsack realizes that he has no idea about the reasons for this statistical rule of thumb, so he fakes an urgent phone call and tells Luzer that he'll get back to him. Hastily, Simsack dials the number for Deviation, who loves the question because he has a great classroom example of the "runs rule."

"Come on over and I'll show you," he says to Simsack. When Hartford arrives, Deviation reaches into his pocket and takes out one of those new dollar coins that pour from post office stamp machines when you don't have the correct change. "Hartford, what is the probability of flipping this coin and getting heads?" Responding to a question for which he actually knows the answer, Simsack says: "That's easy. There's a 50-percent chance."

His mentor then asks Simsack what the probability of getting heads on the second flip might be. Again, Simsack says, "Fifty percent, of course." Going further, Deviation poses the question of one's chances of getting seven heads in a row, and Simsack is stumped. Deviation says immediately that the probability of seven heads in a row is 0.0078. How did he derive that number?

Flipping a coin twice represents two independent events. Most statistics courses include a section on the "multiplication rule," which states that if event A and event B are statistically independent, the probability that event A and B will happen is:

P (A and B) = P (A)  P (B)

This is often called "joint probability." To determine the probability of getting heads on the first flip and heads on the second flip: P (heads first and heads second) = P (heads first) x P (heads second) = (0.5)  (0.5) = 0.25

The multiplication rule can be visualized in a probability tree:

There are four possible outcomes, barring the coin's landing on an edge, to flipping a coin twice. E1 is getting heads on the first flip and on the second flip; therefore, the probability would be one in four, or 25 percent. To calculate the probability of seven heads in a row, a probability tree can represent outcomes for seven flips:

That probability could also be determined by raising 0.5 to the seventh power:

(0.5)7 = 0.0078

If Hartford Simsack--or anyone else--kept flipping a coin over and over from now until his birthday, the probability that he would get seven heads in a row is less than one percent.

This is good, but why seven? Why not eight? Why not six? For runs that indicate instability, various sources give different responses:

Who is right? The majority rule would favor eight. If you use the average, it would be 7.7, but don't forget that it was I who made the list. E-mail me and let me know what you think the correct number is, and I will get back to you in a future column.