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Law of Random Numbers

Date: 12/22/97 at 17:47:25
From: Scott Christie
Subject: Law of random numbers

My question is about the law of random numbers. I've recently been 
debating over the lottery. In Texas we choose 6 numbers between 1 and 
50, so the probability of winning is 1/50x49x48x47x46x45. If I were 
to list every combination of numbers they would all be equally likely 
to be drawn. So, for example, the combination 1, 2, 3, 4, 5, 6 should 
be the same probability as say 5, 10, 18, 33, 45, 50.  

It has been brought to my attention  that the law of random numbers 
changes the odds of picking numbers in order, or close in order. I 
have been told that numbers in order, such as 1,2,3,4,5,6, have a 
smaller probability than numbers that are more random.  Is this true 
and how is it proven?

Date: 12/22/97 at 19:48:42
From: Doctor Tom
Subject: Re: Law of random numbers

Hi Scott,

First off, your probability isn't quite right. The number of
combinations of 6 things from a set of 50 is:


You have to divide by 6! (6 factorial) since the order isn't 

But every combination of numbers is EXACTLY equally likely (assuming
that the lottery isn't rigged). So you have exactly the same chance
of winning if you pick 1,2,3,4,5,6 or 5,10,18,33,45,50. Think about
it this way - if the balls were renumbered, the probabilities should
be the same, right?

In real lotteries, there may be a good reason not to pick things like
1,2,3,4,5,6, however. I don't know about Texas, but in California, if 
there are multiple winners, they split the prize. You'd like to avoid 
that, so what you'd like to do is pick numbers that other folks are 
unlikely to pick. So for that reason, I'd avoid things that I know 
superstitious people would tend to use - 7, 13, and probably all the 
numbers less than 31, since people tend to go for birthdates, etc.

If everyone picks numbers totally at random, or if you are the only 
person playing, any combination is exactly equally likely to win.

But one word of advice: "Don't play the lottery." Again, I don't
know the rules in Texas, but if you win the California lottery,
you're paid in 20 equal yearly installments. So, for example, if you 
won a million dollars, you'd get $50,000 each year for 20 years. But 
money in the future is worth less than money today, and with 
reasonable estimates of interest rates, the 20-year payoff
approximately cuts the current value in half. Also, since when you hit 
the big one you'll be rich, you'll find yourself in the top tax 
category - currently 39% federal, and whatever it is in Texas (in 
California, it's about 11%). So in California, that cuts your winnings 
in half again, so your real prize is only worth about 1/4 of what it 
appears to be.

-Doctor Tom,  The Math Forum
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Associated Topics:
High School Probability

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