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Another way we can be fooled mathematically is on the subject of randomness

Notes On Tricky Use Of Math

By --September 29, 2017

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A bat and ball cost $1.10 in total. The bat costs $1 more than the ball. How much does the ball cost?

Almost everyone who reads this question will have an immediate impulse to answer ‘10 cents.’ I surely did. As Dan Gardner says, “It just looks and feels right. And yet it’s wrong. It’s clearly wrong—if you give it some careful thought—and yet it is perfectly normal to stumble on this test. Almost everyone we ask reports an initial tendency to answer ‘ten cents,’ write psychologists Daniel Kahneman and Shane Frederick. Many people yield to this immediate impulse. People are often content to trust a plausible judgment that quickly comes to mind.” 1

This type of response shows we are quite susceptible to numbers thrown at us by the media, groups seeking funding for a specific cause, lawyers trying to convince a jury, or perhaps some recent event that has shaped our thoughts. 1

The answer is 5 cents.

Here’s another example where we could be mislead by numbers. What is the probability that at least two people in a group of 30 were born on the same day of the year? Perhaps your first inclination is to say that it is a small probability. Once again, I certainly did.

The answer to this apparently simple question is quite surprising and goes against our common sense. The probability is 70.7% and such a large number is truly surprising and much higher than our intuition dictates.

Perhaps even more surprising than this result is the fact that a group of 60 people has a probability of 99.4%, and a group of 80 has a probability of 99.99%. Table 1 provides a more detailed list.2

Table 1- Probability (P) that at least two persons have the same birthday in a group of N persons 2

  • N—P(%)
  • 5 —2.7
  • 10—11.7
  • 20—41.3
  • 30—70.7
  • 40—89.2
  • 50—97.1
  • 60—99.4
  • 80—99.99


Another way we can be fooled mathematically is on the subject of randomness.

Jim Davies uses the following example. Peter Revesz teaches the concept of randomness in a brilliant way. He would ask half of his students to flip a coin 100 times and record the result. He asked the other half to come up with a sequence of heads and tails 100 long that looked random. He told the group to write the two strings of heads and tails on the board and that he would come in a guess which one was actually random. (3)

With one look he could tell which was the true random sequence of heads and tails. Revesz claims that the students can be classified back into their original groups with a surprising degree of accuracy by means of a very simple criterion: In students’ simulated patterns, the longest run of consecutive heads or consecutive tails is almost invariably TOO SHORT relative to that which tends to arise from actual coin tossing. For example, in one case, the real coin tossing sequence has a longest run of eight heads (twice), while the longest run found in the other sequence was only five heads long. 3


Continued below...

Relative Risk Versus Individual Risk

A tool that science and activists use to gain attention is the ‘relative risk’ method. Here’s an example:

“Yearly Stool Test Reduces Colon Cancer Deaths By 33 Percent”

“Yearly Stool Test Reduces Your Chance of Colon Cancer By Less Than 1 Percent”

Both are accurate statements from a very large study (46,551 participants), but one speaks to relative risk while the other covers individual risk reduction. 4

In the group that received annual screening for blood in the stool, 2.6 percent died of colon cancer, while in the group that did not receive annual screening for blood in the stool, 3.4 percent died of colon cancer. The relative risk reduction was 33 percent.

However, the individual risk reduction, e.g., the difference between 2.6 percent and 3.4 percent is 0.8 percent. Therefore, the chance that annual screening for blood in the stool will prevent you from dying of colon cancer is less than 1 percent. Clearly, the headline that suggest 33 percent reduction will get the attention.


D. J. Murphy states, “The medical profession and the media advertise relative risk reduction and not individual risk reduction.” Table 2 provides help in showing the difference between relative risk reduction and individual risk reduction. 4


  1. Dan Gardner, “Risk: The Science and Politics of Fear, (Toronto, McClelland & Stewart, 2008), 35
  2. Robert B. Banks, Towing Icebergs, Falling Dominoes, and Other Adventures in Applied Mathematics, (Princeton University Press, 1998)
  3. Jim Davies, Riveted, (New York, Palgave Macmillan, 2014)
  4. D. J. Murphy, “Honest Medicine,” The Atlantic Monthly Press, 1995

Jack Dini -- Bio and Archives | Comments

Jack Dini is author of Challenging Environmental Mythology.  He has also written for American Council on Science and Health, Environment & Climate News, and Hawaii Reporter.

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