## More On Opposites

The article two days ago dealt with seeing the study of memory as the study of forgetting and how that kind of insight transfers to other things in life. The idea of looking at the other side is quite common in mathematics and people often reach very wrong answers by using intuition.

A common example is this bet. There are 30 people in the room. If you give me 3 to 1 odds I will bet there will be at least one pair that celebrates their birthday on the same day of the year.  I put up \$100, and you offer to pay me \$300 if there is a match.  Most people look at that and see 30 people 365 days and think 1 chance in 12. These people are the victims.

One in 12 is not close. The problem is in seeing it from the positive side.  Think about it from the negative side and you get this.

The chance of the second person matching the first is 1 in 365 and the chances of them not matching is 364 in 365. The chance of the third matching either of the other two is 2 in  365 and the odds of not is 363 in 365. And so on to the 30th person who is 30 in 365 matching someone and 335 not. About the 1 in 12 we had before but only for the last person.  It might have matched earlier so intuition is wrong.

Knowing the positive odds, the 1 in 365 and all the others tell us nothing. The negative side is useful because we can combine the odds by multiplication. The odds of no matches is 365×364/365×363/365 and so on to 336/365.

Now we find that the probability of not matching any two is around 29% or roughly 2.5 to 1 in favour.  Offering to pay \$300 on a \$100 bet when you should offer to pay only \$29 implies that the bettor stands to make 10 times what they should.

If you work it out with a spreadsheet you will see it is a 50-50 bet with 23 people.  Think it through with your family or your office.

With 100 people there will be a match 9,999,997 times in 10,000,000.

A little different if we adjust for leap year babies.

Be very careful with statistics, probabilities and along the same line, with compound growth. Intuition is the product of our ancient skill set. Compound growth, statistics and probability theory were not factors early humans considered so we don’t yet have intuitive strength in the area.

Intuition is not always right and sometimes you can defeat your instinctive answer by looking at it from the other side.

Don Shaughnessy arranges life insurance for people who understand the value of a life insured estate. He can be reached at The Protectors Group, a large insurance, employee benefits, and investment agency in Peterborough, Ontario.  In previous careers, he has been a partner in a large international public accounting firm, CEO of a software start-up, a partner in an energy management system importer, and briefly in the restaurant business.