by Tom Rogers, Twitter Link
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The Awesome Power of Twenty Questions

It's a common classroom game for gradeschoolers and yet it contains a profoundly powerful problem solving strategy which can be used to de-bug software, troubleshoot equipment and solve problems in business and industry. In the game one child picks an object and everyone else has 20 questions to identify it. At first the children guess specific items but they soon realize they need to eliminate entire categories or risk using up their 20 questions before finding the answer. 

The principles of twenty questions are frequently used in the business world to conduct computerized searches of massive data bases. These are called a binary searches and are one of the fastest search methods available. To conduct binary searches, data must be sorted in order or alphabetized. The computer determines which half of the list contains the item. The half containing the item is divided in half again and the process repeated until the item is found or the list can no longer be divided. 

Just how powerful is 20 questions? The correct alternative can be identified from among 220 or 1,048,576 different alternatives assuming that a "no" has the same weight as a "yes" answer. In other words, either answer eliminates half the alternatives on each question. For example, consider the question, "is the object in this half of the room." A yes or no question will generally eliminate half of the objects in the room. (The exceptions are objects which are in both halves such as the air.)

Is weighting yes and no answers equally really the best technique? At first glance it seems like weighting the yes so that it eliminates 80% of the alternatives would be even better. True, a no answer would only eliminate 20% of the alternatives but surely we could ask the questions so that we would get yes answers at least 50% of the time. 

Each yes will leave 20% of the questions remaining. In other words a single yes could identify the correct choice from a group of five alternatives with only one question as follows:
A1 = 1/(0.2)1
= 5

where: A1 is the number of possible alternatives or the power for a single question.

For 10 yes answers the number of alternatives would be:
A10  = 1/(0.2)10
= 9,765,625

If we add another 10 no answers, the number of alternatives rises as follows:
A20  = (1/(0.2)10)(1/(0.8)10)

This is impressive but, unfortunately not attainable. The probability of getting a yes will not be 50%. It will be 100% - 80% or only 20%. The probability of getting a no will be 80%. Consider an example: If there were one correct choice among five alternatives and we asked if a particular one was correct, a yes answer would eliminate 80% of the alternatives. Unfortunately, the alternative we inquire about has to be the correct one to get a yes. Hence, the probability of a getting yes answer is only 1/5 or 20%.

The correct way to calculate the number of alternatives for 20 questions with a yes eliminating 80% of the alternatives is as follows:
A = (0.2)(-20*.2)(0.8)(20*.8)
= 22,204

In reality, trying to eliminate 80% of the alternatives with each yes question is a very poor strategy.

We can write a general purpose expression for calculating the alternative using various weighting for a yes answer as follows:
= (Py)(-20*Py)(Pn)(-20*Pn)


Py = the probability of getting a yes answer

Pn = the probability of getting a no answer

Pn = 1 - Py

Note that the above equation will only give meaningful values when Py is a factor of 1/20. The exponent for either Py or Pn represents the number of questions answered yes and no respectively. Obviously it's impossible to have a fraction of a question answered yes or no. Hence, both Py and Pn have to be factors of 1/20 to insure that both exponents represent an integer number of questions.

Figure 1 illustrates the power of twenty questions for various weightings of a yes answer. Power is defined as the number of alternatives including the correct one which can exist and still be able to identify the correct one with only 20 questions. The weighting of a yes answer indicates the fraction of the alternatives which can be eliminated by a yes answer. Obviously, a 0.5 or 50% weighting on yes questions is the optimum strategy. This means that either a yes or no answer can on average eliminate half of the alternatives. We say on average since it is not possible to eliminate exactly half of the choices when there is an odd number of alternatives. Note, that wasting a question cuts the power in half.

The principles of twenty questions can be used in any trouble shooting or problem solving situation where the problem is associated with a single element of a complex system. Problem solvers should avoid focusing on the cause and  instead ask which elements of the system can be eliminated as causes. Data collection and experiments should attempt to eliminate half of the alternatives with each effort. 

Element should only be eliminated based on data. No element should be omitted because it is too new, traditional, high tech or expensive to be the problem.

Typical data collection techniques often will not eliminate elements. For example a full gas gauge would not eliminate the fuel system as a reason why a car won't start. The fuel pump might be broken. A great deal of creativity is often needed in collecting data. However, low tech methods such as looking at the equipment should not be ignored even when no one knows what to look for. After all Columbus discovered America by looking for India.

When most of the elements in a system have been eliminated its time to develop a hypothesis about what caused the problem. Using this hypothesis, list possible causes and test them until a solution is found. The chances of success are much higher when working with a limited number of elements rather than the entire system

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