Playing the Lottery One Time or Weekly

Date: 06/09/2008 at 16:22:33
From: Jennifer
Subject: lottery: many tickets in 1draw vs 1ticket in many draws

Are the odds higher if you purchase 10 tickets for one lottery style
random number draw, or 1 ticket for each of 10 separate draws? (Say
for example if you spend $52 for a single draw rather than $1 a week 
for a years worth of weekly draws.)

There is a similar question in the archives which clarifies that odds
in a random number draw are dramatically increased if you have more
than one ticket in a single drawing, but not how this compares to
having a single ticket for a string of separate drawings.

It appears to me that the first scenario should have better odds, 
since for each wrong ticket in your set, you have eliminated one 
chance of picking a loser, and so your next ticket has better odds,
whereas if you have one ticket in each separate draw, you are always
starting at the highest odds each time with each ticket.  

If I imagine the scenario as buying 1 ticket for a 1:2 drawing on 2
occasions, I have a 1:2 chance of winning/losing in each drawing, and
no guarantee of winning in either.  However if I buy 2 tickets for the
first drawing, I have a 2:2 chance of winning--i.e. 100%.

On the other hand with a lottery it seems intuitive to spread your
bets or "not put all your eggs in one basket" so it feels like you are
getting more chances if you are "in" more games.  Which is it and how
is that calculated?

Date: 06/09/2008 at 20:23:59
From: Doctor Tom
Subject: Re: lottery: many tickets in 1draw vs 1ticket in many draws

Hi Jennifer,

I assume that when you buy tickets in a single game that you're always
smart enough to choose all different tickets.

So let me answer this question:  If N > 1, do you have a better chance
of winning by getting N tickets in one game or 1 ticket in N games,
where we assume also that N is at most as large as the largest number
of possible tickets.  In other words, you don't buy 101 tickets in a
lottery with only 100 outcomes.

The answer is, as you suspected, that you have a better chance of a
win with N tickets in one game.

This may seem slightly counter-intuitive, but the reason is that
although you don't win as often in the "play once N times" scenario,
when you play once N times, you'll sometimes win more than once.

The AVERAGE number of wins is the same in both cases.  If there are M
possible choices of tickets and you make N bets, your expected number
of wins will be N/M in both cases.

Let's look at your simple example:  M=2 and N=2.

If you buy both tickets in a lottery, one ticket always wins, but the
other always loses, so you win 1 time.

If you buy 1 ticket in each of two lotteries, there are three
possibilities:

  you lose both (payoff = 0)
  you win one (payoff = 1)
  you win both (payoff = 2)

Winning exactly once occurs twice as often (you could win the first
and lose the second, or vice-versa).

Your expected payoff in this case is:

  1/4(0) + (2/4)1 + (1/4)2 = 1 -- same average payoff.

And this works for any sized lottery.  I'll do it for three tickets. 
If you play three times, here are the ways the lottery could come out,
with W being a win and L a loss.  Since we're betting on 1 of 3, the L
happens 2/3 of the time for any game and the W 1/3 of the time:

  WWW (1/3)(1/3)(1/3) = 1/27

  WWL (1/3)(1/3)(2/3) = 2/27
  WLW (1/3)(2/3)(1/3) = 2/27
  LWW (2/3)(1/3)(1/3) = 2/27

  WLL (1/3)(2/3)(2/3) = 4/27
  LWL (2/3)(1/3)(2/3) = 4/27
  LLW (2/3)(2/3)(1/3) = 4/27

  LLL (2/3)(2/3)(2/3) = 8/27

I've broken them into four groups, with 3, 2, 1 and 0 wins.  The odds
of each of the results occurring are shown to the right of the W/L
pattern.

1 time in 27, the payoff will be 3, 6 times in 27, it will be 2, 12
times in 27 it will be 1, and 8 times in 27, zero.  Your expected
payoff will thus be:

  (1/27)3 + (6/27)2 + (12/27)1 + (8/27)0 
  (3 + 12 + 12 + 0)/27 = 27/27 = 1.

Same payoff (on average) as betting all three lottery numbers in the
same game.

But note that you ALWAYS "win" if you bet all three numbers, and you
only win 19 times out of 27 if you bet once in each game.  But if you
bet all three numbers in one game, although you always win, you always
win once, and sometimes with the one in each game you win twice or
even three times.

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