Whoever told you to mail me a free copy of *Miller-McCune* certainly had their insider information right. I’ve spent most of the day reading almost all the great articles. I love your compassion, optimism, realism, worldwide perspective and data-based and solution-based approach. I intend to subscribe tomorrow.

I do have a disappointment, however, and I think I understand your reasoning. You do not wish to emphasize the dangers of population growth because it does not lead to any cheerful solutions.

Colleen Shaddox points out (“Simply Rwandan,” March-April) that Rwanda, the size of Vermont, has a “rapidly growing population of 10 million,” and also that “Rwanda has always been a country of large families.” Almost certainly the population will exceed 20 million before many decades, right? It is not at all realistic to assume that most of the additional 10 million will be bankers, tourist guides, software programmers and high-tech technicians. We can assume that there will therefore be intense pressures for land — for cows and farming. Yes, I appreciate that all the articles end up on an upbeat note. But wouldn’t effective steps toward family planning make Rwanda’s future much brighter?

I live in Guatemala, Central America, and the resistance to family planning is very similar. In my lifetime, the population has doubled twice and will likely double again to about 25 million in the next 30 years. Probably not coincidentally, we have had a long, bloody, evil and horrible civil war. We are now beset by drug running, organized crime, family violence and violence against women, environmental degradation and massive under-employment. Would family planning have completely prevented the suffering? Of course not. But population growth will eventually have to end. Humans can make the choices — or let tragedy make the choices for us.

**Paul E. Munsell, Ph.D.***Guatemala City, Guatemala*

**For Those of You Who Paid Attention in Statistics Class…**

Can we ever prevent the imprisonment of innocent people?

Following up on Steve Weinberg’s article (“Innocent Until Reported Guilty,” October 2008) and the subsequent commentary, let me offer a sad but sober dose of mathematical reality. The conclusion is that so long as only a very small minority of people commit crimes and the criminal justice system is fair (“fair” meaning that all people are equally subject to investigation) there will always be a very large proportion of innocent people convicted.

Now the supporting argument, made by way of an example:

Suppose there are 10,000 true criminals in the United States annually. I don’t know how many there really are, but let’s assume 10,000 for the example.

Second, let’s assume that the criminal justice system is 99.9 percent accurate. By “system” I mean the entire system, starting with investigation and prosecution and ending with punishment. I know that 99.9 percent may be Pollyannaish, but, again, let’s accept it for the example. This means that the probability of a guilty person being caught and successfully tried, convicted and punished is 99.9 percent and the probability of an innocent person being convicted is but 0.1 percent.

Now let us ask and answer the key question: If a person is found guilty of a crime, what is the probability that s/he is guilty?

This probability is a ratio that has, in its numerator the number of guilty people successfully punished = .999 x 10,000 = 9,990. In the denominator is the number of guilty people being punished plus the number of innocent ones being punished. We already have the guilty part of this (9,990). The innocent part is .001 times the number of innocent people in the U.S., or .001 x 300,000,000 = 300,000.

So the answer to the question is:

9,990/(9,990+300,000) = 9,990/309,990 = 3%

Or, expressed another way, 97 percent of those in prison, under the circumstances of this example, would be innocent. Of course if the true number of criminals is 100,000, then the proportion of innocents is “only” 70 percent. It is amazing that so few horror stories are being told.

**Howard Wainer***Professor of Statistics
The Wharton School
of the University of Pennsylvania*

**A question from the editor:** We’d like to double-check some of your reasoning with you. You create a fraction with the number of people successfully punished in your example in the numerator (in this case, 9,990) and in the denominator, you use 300,000,000 as the number of innocent people in the U.S.

We were wondering, for your example to be valid, wouldn’t you have to place a number of “charged” innocent people in the denominator, and not the entire U.S. population?

**A response from Wainer:** Thank you for taking the time to read and think about my example. No, the denominator is as I have specified it. The figure 99.9 percent represents the probability of getting it right of the whole process — this means initial investigation, charging, prosecuting, convicting and imprisoning.

So it assumes that at the beginning of any investigation, everyone is under consideration (although a large proportion may be eliminated quickly). This assumption may not be true — it may be that some groups of people (the usual suspects) are always considered, and some never are. I was proceeding under the democratic assumption that initially at least we are all equal under the law.

Note, by the way, that the same arithmetic is informative in evaluating medical testing. Each year in the U.S., 186,000 women are diagnosed, correctly, with breast cancer. Mammograms identify breast cancers correctly 85 percent of the time. But 33.5 million women each year have a mammogram and when there is no cancer it only identifies such with 90 percent accuracy. Thus if you have a mammogram and it results in a positive (you have cancer) result, the probability that you have cancer is:

186,000/(186,000+3.35 million) = 4%

So if you have a mammogram and it says you are cancer free, believe it. If it says you have cancer, don’t believe it.

The only way to fix this matches your question — reduce the denominator. Women less than 50 (probably less than 60) without family history of cancer should not have mammograms.

**An editor’s challenge:** We still suspect that professor Wainer knows his statistics cold but scored less well in Assumption Making 101. If you know we’re right or wrong, go online at Miller-McCune.com and tell us why. Our comments section awaits your brilliance.

Miller-McCune *welcomes letters to the editor, sent via e-mail to theeditor@miller-mccune.com; via the comment sections of our Web site, Miller-McCune.com; or by standard mail to The Editor, 804 Anacapa St., Santa Barbara, CA 93101. Letters published in the print magazine may be condensed for space reasons.*

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