Marilyn vos Savant has the highest reported IQ in the world and writes a weekly column where she answers questions from the great unwashed. Often time, the questions are simple enough that a person with average intelligence could answer them. Sometimes, though, she does bobble one and give an obviously wrong answer. (I guess your editor can’t question your work if you’re the smartest person in the world.)

Let’s take a look at this recent question.

I manage a drug-testing program for an organization with 400 employees. Every three months, a random-number generator selects 100 names for testing. Afterward, these names go back into the selection pool. Obviously, the probability of an employee being chosen in one quarter is 25 percent. But what’s the likelihood of being chosen over the course of a year?
Jerry Haskins, Vicksburg, Miss.
 
Marilyn responds:
The probability remains 25 percent, despite the repeated testing. One might think that as the number of tests grows, the likelihood of being chosen increases, but as long as the size of the pool remains the same, so does the probability. Goes against your intuition, doesn’t it?

Yes, it certainly does go against my intuition.  There’s a great reason for this – the answer is wrong.

Is the answer counter-intuitive?

Before we actually analyze in any detail, ask yourself if this makes sense.  Do you really think a person has a 25% chance of being chosen over the course of a year, regardless of the number of tests?  So if the company tests 1 time per year or 700 times per year (arrival, lunch break, and right before you leave every day) John Q. Public on the assembly line has a 25% chance of being picked at any time during the course of a year?

Give Marilyn a point for correctly suggesting that her answer is counter-intuitive.

Walk through the 4 tests

OK, let’s walk through the selections in each of the quarters.

  • Quarter 1: Of the 400 employees, 100 are selected and 300 are not.  At this point, we’ve broken the single group of employees into two sub-groups – those who have been selected and those who have not.
  • Quarter 2:  100 of the 400 employees are selected again.  If the sampling is truly random, 25% of each sub-group will be selected.  This means that 25% of the 100 employees (25) who were selected in the 1st quarter will be selected again, and 25% of the 300 employees (75) who were not selected in the 1st quarter will be selected.  The “selected at least once” sub-group now grows to 175 while the “never selected” subgroup shrinks to 225.  From this point on, we’ll focus on the “never selected” group.
  • Quarter 3:  The “never selected” group drops to 169.
  • Quarter 4:  The “never selected” group drops to 127.

At the end of the year, 127 of the 400 employees (31.75%) have completely avoided the testing, while 273 (68.25%) have been selected at least once.  1 or 2 people would have been selected all four times.

Show me the math

As is often the case with probability, the easiest way to attack this is to computer the odds of the opposite circumstance and subtract this from 100%.  The odds of being selected one of more times would involve computing the odds of being selected once, twice, three times, or four times and then adding the results.

Alternately, we can easily calculated the odds of never being selected, and just subtract this from 100% to arrive at the likelihood of being selected at least once.

The odds of avoiding testing in any quarter is 75%.  Thus, we simple raise .75 to the power of 4 (.75^4) to arrive at the odds of never being selected for testing – .3164, or 31.64%.  Thus the odds of being selected at least once is 68.36%.  This differs slightly from the result above because the 68.25% involved some rounding (since we must use whole people and not fractions).

The moral of the story?

Don’t place too much trust (or dis-trust) in the messenger.  Pay attention to the actual message.

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Kosmo is the founder of The Soap Boxers and writes on a variety of topics. Many of his short stories have been collected into Kindle books.

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