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# How I Avoided the Doghouse Today

Whew! I got through another year without being marched toward the domestic gallows: Today is my wife’s birthday, and I remembered it. Safe for another year!

I don’t think she’d want me to advertise the number associated with her birthday (let’s just say she’s been 21 more than once now), but there is another number worth considering because it is illustrative of the fact that most people guess numbers wrong.

Before I get to that number, consider another one: I read recently that the average American thinks that 25 percent of his fellow countrymen are homosexual. The real figure is about 2 percent. How is it that so many people get the number so wrong? Probably because the media incessantly talk about such things as same-sex marriage and the supposed rights that homosexual persons are deprived of. If you keep hearing about certain topics, and if you are indisposed to doing any research on your own, then, at least on sociological-type matters, you give some things entirely too much emphasis.

The number I have in mind for this blog post isn’t related to homosexuality. It’s related to birthdays. It shows how off the mark people can be when trying to do any thinking—or even guesstimating—that involves math. Here is the question:

If you have a group of people—say people attending a party together—how many people need to be present to have the probability be at least 50 percent that two of the people—any two of them—have the same birthday (month and day, not year)?

Not taking into account leap years, one thing we can say is that if we have 366 people, at least two of them must have the same birthday, since there are only 365 days in a year. Obviously, we won’t need that many at the party to have a high likelihood of two people sharing a birthday. If we have 300 people at the party, we probably can sense that it’s more likely than not that at least two of them share a birthday. But how few do we need to have the probability break the 50 percent mark?

Most people don’t give the matter much thought. They arrive at an answer by “feel,” not by mathematical logic. They see the figure of 50 percent in the question and guess that we would need 183 people, 183 being the first integer that’s over half of 365. “Good enough for government work,” they say, satisfied with their answer.

But their answer isn’t remotely good enough even for government work, because the correct answer is 23. If we have just 23 people in the room, the likelihood of two of them sharing a birthday is more than 50 percent.

You may not believe me, so let me explain how the answer can be derived. I’ll show you how to do it but won’t work out the calculation all the way. You can do that on your own, if you’re so inclined.

The easiest way to arrive at the solution is to work the problem upside down: How many people can we add while keeping the odds that no two of them share a birthday at less than 50 percent?

Let’s say you have just two people in the room. The odds that they don’t share a birthday are 364/365, or .997. Whatever the birthday of the first person might be, the next person can have a birthday that falls only on one of the other 364 days of the year, so his odds are 364 in 365, or 364/365.

If we add a third person, there now are only 363 days to choose from, giving us 363/365 for him. To find the overall odds so far, we multiply the figures together: 364/365 x 363/365, or .992. With a fourth person we multiply by another fraction and get 364/365 x 363/365 x 362/365, or .983.

By the time we get to person number 23, we get 343/365 as his fraction. When that is multiplied by all the previous fractions, we get .493. This is the probability (49.3 perent) that all 23 people have different birthdays. Subtract that number from 100 percent and you’re left with 50.7 percent, the probability that at least two of the people share a birthday.

The next time you go to a party, you can stump the attendees with this puzzle. Of course, most of them won’t believe your answer. It just doesn’t “make sense,” but it doesn’t “make sense” to lots of people because they have no real idea about how one can derive an answer.

It’s that way with religion. Many people are satisfied with a simple but wrong answer. They say the Catholic faith is “too complicated”: too many doctrines, too many rules, too many customs. They opt for a stripped down religion. But our instincts should tell us that a religion the entire contents of which can be written on a 3×5 card can’t possibly be the true religion. The world is complex, and only a big religion is big enough for it.

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