Lottery Strategy and Odds of Winning

Date: 04/27/2000 at 16:25:58
From: Martin Smith
Subject: Odds of winning vs. Odds of Losing

My friend maintains that my odds of winning in the UK National Lottery 
are the same whether I choose the same numbers each time or whether I 
choose different numbers. I maintain that my odds of losing are 
greater if I change my numbers than if I keep the same ones. The 
theory is:

FACT: The odds of winning are approximately 13 million to one. So 
according to my friend:

1. No set of numbers is any more likely to win than any other.
2. The balls and machines have no memory, therefore each draw is a new 
   one.
3. If you entered 13 million times you would win.

BUT I maintain:

1. If you change your numbers you may have a winning set the week 
   before that set wins. In other words, you could enter 13 million 
   draws and get the next week's winning numbers every time and still 
   lose every time.

2: If you enter 13 million times with the same numbers, you will win 
   because your number is bound to turn up for one of them.

I know this is not exactly the case as it is perfectly possible to 
have the same numbers more than once; however, I believe this will 
affect both scenarios in the same way.

So who is correct?

Date: 04/28/2000 at 15:59:30
From: Doctor TWE
Subject: Re: Odds of winning Vs Odds of Losing

Hi Martin - thanks for writing to Dr. Math.

Assuming that we are talking about playing the numbers on different 
weeks (as opposed to buying multiple tickets for the same week), and 
assuming that the generating machine is fair, then your friend is 
right. No set of numbers is any more likely to win than any other.
This is correct if the device is fair. In probability, we usually 
assume a fair random number generator, but in the "real world" this 
may not be the case. Sometimes machines are deliberately rigged;
sometimes they just aren't perfectly designed.

2. The balls and machines have no memory, therefore each draw is a 
   new one.

This is absolutely correct. Each week's drawing is an independent 
event, not affected by past or future drawings.

3. If you entered 13 million times you would win.

Not necessarily. On the _average_ you should win once in 13 million 
drawings, but they may repeat numbers that aren't yours and not pick 
yours. If it's a fair machine, in 13 million million drawings, you 
should win about 1 million times.

This is like saying that if you roll a die 6 times, you'd get a 6 
exactly once. This is not the case. On the average, 1/6 of the rolls 
will be 1's, 1/6 of the rolls will be 2's, etc., but we could get two 
4's and no 6. Likewise, we could get three 6's but no 2's. In enough 
rolls, it will average out. Likewise, in enough lotteries (billions of 
them), it will average out.

>1. If you change your numbers you may have a winning set the week 
    before that set wins. In other words, you could enter 13 million 
    draws and get the next weeks winning numbers every time and still 
    lose every time.

Say your number is 1-2-3-4-5. It is equally likely that every week, 
the winning combination will be 6-7-8-9-10. Each week you will miss it 
by "that much".


>2. If you enter 13 million times with the same numbers, you will win 
    because your number is bound to turn up for one of them.

See what I said about your friend's point 3. Not every number will 
come up in those 13 million drawings.

An important concept that you're missing is the idea of replacement. 
If, once drawn, the numbers are removed from the pool and cannot be 
picked ever again, then sooner or later you would win in 13 million 
drawings. If you stuck with the same number, sometime in those 13 
million drawings your number would come up. If you changed numbers 
each week (we'll assume you only choose from those that haven't yet 
been picked), sooner or later you'll pick the winning one. If, by the 
13 millionth drawing, you hadn't picked the winner yet, you'd know 
which one was left.

This would be like guessing which card will be picked from a deck and 
placed on a table. Your chances are 1 in 52 when the first card is 
picked. Say that it's the 5 of clubs. If you're wrong, the card is 
placed on the table (NOT reshuffled into the deck) and you try to 
guess the next card. If you haven't guessed the card in the first 51 
draws, you can figure out which card is left and guarantee that you'll 
"guess" right on the 52nd draw.

But the lottery is not like that. Each time a combination is picked, 
it is "reshuffled" into the mix and can be picked again in any of the 
following weeks. The reason that we don't seem to "see" this is 
because of the small sample size relative to the pool of possible 
combinations. If we COULD play millions of millions of lotteries (we 
can simulate this with computers), we'd see repetitions, and we'd see 
that the odds of winning are the same for either strategy.

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum
  http://mathforum.org/dr.math/