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Statistics Hacks by Bruce Frey

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Hack #46. Design Your Own Bar Bet

With a few calculations, and perhaps some spreadsheet software, you can figure the probabilities associated with all sorts of "spontaneous" friendly wagers.

Several of the statistics hacks elsewhere in this chapter use decks of cards [Hack #42] or dice [Hack #43] as props to demonstrate how some seemingly rare and unusual outcomes are fairly common. As someone who's interested in educating the world on statistical principles, you no doubt will wish to use these teaching examples to impress and instruct others. Hey, if you happen to win a little money along the way, that's just one of the benefits of a teacher's life.

But there's no need to rely on the specific examples provided here, or even to carry cards and dice around (though, knowing you the way I think I do, you might have plenty of other reasons to carry cards and dice around). Here are a couple of basic principles you can use to make up your own bar bet with any known distribution of data, such as the alphabet, numbers from 1 to 100, and so on:

Principle 1

An unlikely event increases in likelihood if there are repeated opportunities for it to occur.

Principle 2

If there are a large number of possible events, the chance of any specific event occurring seems small.

The rest of this hack will show you how to use these principles to your advantage in your own custom-made bar bets.

Principle 1

The probability of any given event occurring is equal to the number of outcomes, which equal the event divided by the number of possible outcomes. For example, what are the chances that you and I were born in the same month? Pretending for a second that births are distributed equally across all months, the probability is 1/12. There is only one outcome that counts as a match (your birth month), and there are 12 possible outcomes (the 12 months of the year).

What about the probability that any one of two people reading this book has the same birth month as me? Intuitively, that should be a bit more likely than 1 out of 12. The formula to figure this out is not quite as simple as one would like, unfortunately. It is not 1/12 times itself, for example. That would produce a smaller probability than we began with (i.e., 1/24). Nor is the formula 1/12 + 1/12. Though 2/12 seems to have promise as the right answer—because it is bigger than 1/12, indicating a greater likelihood than before—these sorts of probabilities are not additive. To prove to yourself that simply adding the two fractions together won't work, imagine that you had 12 people in the problem. The chance of finding a match with my birth month among the 12 is obviously not 12/12, because that would guarantee a match.

The actual formula for computing the chances of an event occurring across multiple opportunities is based on the notion of taking the proportional chance that an event will not happen and multiplying that proportion by itself for each additional "roll of the dice." At the conclusion of that process, subtracting the result from 1.0 should give the chance that the event will happen.

This formula has a theoretical appeal because it is logically equivalent to the more intuitive methods (it uses the same information). It is appealing mathematically, too, because the final estimate is bigger than the value associated with a single occurrence, which is what our intuition believes ought to be the case. Think about it this way: how many times will it not happen, and among those times, how many times will it not happen on the next occurrence?

Here's the equation to compute the probability that someone among two readers will have the same birth month as I do:

image with no caption

Principle 2

To get someone to accept a wager or to amaze an audience with the occurrence of any given outcome, the likelihood must sound small. So, wagers or magic tricks having to do with the 365 days in a year, or the 52 cards in a deck, or all the possible phone numbers in a phone book are more effective and astounding because those numbers seem big in comparison to the number of winning outcomes (e.g., one).

The chance of any unlikely event occurring on any single event is indeed small, so the intuitive belief expressed in this principle is correct. As we have seen, though, the chances of the event occurring increases if you get more than one shot at it, and it can increase rapidly.

Rolling Your Own Bar Bet

Let's walk through the steps that verify my advantage for a couple of wagers I just made up.

Letters of the alphabet

For this wager, I'll pick five letters of the alphabet. I bet that if I choose six people and ask them to randomly pick any single letter, one or more of them will match one of my five letters. Here's how the bet plays out:

Number of possible choices

There are 26 letters in the alphabet.

Probability of a single attempt failing

There are 21 out of 26 possibilities that are not matches: 21/26 = .808.

Number of attempts

6

Probability of all 6 attempts failing

.8086 = .278

Probability of something other than the previous options occurring

1 - .278 = .722

The chance of my winning this bet is 72 percent.

Pick a number, any number

This time, I'll pick 10 numbers from 1 to 100. I bet that if I choose 10 people and ask them to randomly pick any single number from 1 to 100, one or more of them will match one of my ten numbers. Here's how this one works out:

Number of possible choices

There are 100 numbers to choose from.

Probability of a single attempt failing

There are 90 out of 100 possibilities that are not matches: 90/100 = .90.

Number of attempts

10

Probability of all 10 attempts failing

9010 = .349

Probability of something other than the previous options occurring

1 - .349 = .651

The chance of my winning this bet is 65 percent.

On your own

Copy the steps and calculations just shown to develop your own original party tricks. None of these demonstrations require any props, just a willing and honest volunteer.

Notice that the calculations are based on people randomly picking numbers. In reality, of course, people will not pick a letter or number that they have just heard someone else pick. In other words, their choices will not be independent of other choices. If the choices are made based on the knowledge that previous answers are not correct, this helps your odds a little bit. For example, on the 10-out-of-100 numbers wager, if there is no chance that the 10 people will choose a number that has already been chosen, your chances of getting a match go from 65 percent to 67 percent.

Make Sure the Sucker Isn't You!

It is fun to play with others, but you never know when you will get caught in someone else's clever statistics trap. For instance, remember that 1-out-of-12 chance that you have the same birth month as me? I fooled you! I was born in February. There are fewer days in that month than the others, so your chances of being born in that month are actually less than 1 out of 12. There are 28.25 days in February (an occasional February 29 accounts for the .25) and 365.25 days in the year (the occasional Leap Year accounted for again). The chance that you were born in the same month as me is 28.25/365.25, or 7.73 percent, not the 8.33 percent that is 1 out of 12.

So, you are less likely to have the same birth month as me. Come to think of it, the records of my birth, my birth certificate, and so on were lost in a fire many years ago. So, the original data about my birth is now missing.

For all I know, I might not even be born yet!

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