10 Coins

Lesson 3

If you flipped 10 coins, how many heads would you expect? With no tricks, and standard coins most people would say 5 heads, and that would be correct. Each coin of the ten coins has a 50% chance of being a head, so the expected value for flipping 10 coins is 5 heads (10 coins x .50 heads/coin = 5 heads).

But would it surprise you, if 6 heads occurred? Most people with a little thought would realize that 6 heads is a reasonable possibility and would not be a surprise. Even though it is not the expected or the most common outcome, 6 heads is reasonable and likely.

Now, what would your thoughts be, if 10 heads occurred? Most people would be a little hesitant, but would realize it could happen, but not very often. Note that 10 heads is possible, but unlikely.

I have conducted this little survey with many people, and, generally, it is like the above story. Most people have a natural sense of how heads will occur, though most have not figured or know the real probabilities. This is a great testimony to the experience and the sense that people have about the nature of probability. The real probabilities for heads to occur are 24.6% of the time for 5 heads, 20.5% of the time for 6 heads and 0.10% of the time for 10 heads. Or, it can be said that 5 heads will occur about one-fourth of the time, 6 heads about one-fifth of the time, and 10 heads about 1 time out of one thousand in an honest system.

This simple idea of probability gives us a standard to use to discuss events or statistics. For instance, when an event occurs, we can compare it to the probability of the number of heads in the ten coins example. In data and statistical studies, some events will occur just the way we expect, like 5 heads. And some events will occur almost the way we expect, like 6 heads. Six heads is different, but still expectable and nothing to be surprised about. On the other hand, if an event occurs that is unlikely, like 10 heads in 10 coins, we can understand and respond to this rare occasion, if appropriate.

So, when an apparent difference is found in data, the correct question to ask is how close is the result to what we expect. Is it like 6 heads or 10 heads? Six heads is nothing unusual, like dog bites man, and only a little different from the 5 heads we expected. That difference is just noise, an insignificant difference from 5 heads in 10 coin flips. Ten heads is truly unusual and worthy of note, because it is so rare, and a story to report like man bites dog. And, of course, if it happens too frequently or in a special way, it is a signal of some true difference.

Which Are You

Looking At?

Although this ten coin example does not perfectly represent all applications from the real world, it is a simple example that most people understand from their own experiences. In the next lesson, we will cover how the control chart helps separate signal from noise.

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