Lottery Probability

Date: 01/23/98 at 21:08:56
From: marco and diane
Subject: Lottery probability

We've been having a heated discussion about the probabilty of picking 
winning lottery numbers. Let's say that Monday's winning numbers were 
1,2,3,4,5, and 6. Is there or is there not the same probability that 
those winning numbers will come up again? I mean isn't there the same 
chance that any random numbers or those exact same numbers will be 
repeated 3 days later? (The numbers to choose from are 1-46.)  In 
other words what is the probability that Thursday's numbers will be 
the same as Monday's numbers? (Diane's question.)

I know that the chance of those numbers coming up on any given day is 
1 in 10 million. Now if you play the same numbers as the previous 
winning numbers, doesn't the probability change?  (Marco's question)

Date: 01/23/98 at 22:31:23
From: Doctor Budrow
Subject: Re: Lottery probability

Dear Marco and Diane,

There are two distinct questions here so you are probably arguing 
about different probabilities. Often the hardest thing in probability 
is clarifying exactly what the question is.  

The first question is: what is the probability of one set of six 
numbers being drawn at random if you know the outcome of a previous 
draw?  

The other is: what is the probability of the same set of six numbers 
being drawn twice in a row?  

An equivalent question (but with a simpler example) to the first one 
is: what is the probability of heads on the toss of a coin knowing 
that the previous flip was also heads?   

An equivalent question to the second one is: what is the probability 
of flipping two heads in a row?  

In the case of the coin, the chance of flipping heads knowing the 
previous flip was heads is 1/2, if the coin is fair. It would also be 
1/2 if the previous flip were tails. The chance of flipping two heads 
in a row is 1/4, as is the chance of flipping a heads then a tails or 
a tails then a heads or a tails then a tails.  

The difference between the two questions is that in one case you are 
stating that each flip of a fair coin is independent of the previous 
flip, and in the other you are asking what the probability of a 
specific pair of outcomes is.  

In the case of the lottery, Marco is asking what the chance of winning 
with the same numbers that were played the week before is. This is the 
same as the chance of flipping a coin and getting heads when you know 
the previous flip was also heads, which is 1/2. So, yes, he has one 
chance in 10 million of winning with the same numbers. But if you 
choose a set of six numbers, the chance of that set being drawn twice 
in a row is 1 in 10 million squared, a very small chance indeed.  

One more way to look at this question is to ask: if I examine all the 
winning combinations drawn so far, what is the probability that at 
least one combination has been repeated?

-Doctor Budrow,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 01/24/98 at 00:00:54
From: Doctor Bill
Subject: Re: Lottery probability

Marco and Diane,

Suppose you flip a penny one time. What is the probability of getting 
a head? It's 1/2, right? Suppose you flip the penny a second time. 
What is the probability of getting a head? It's still 1/2, isn't it? 
Does the outcome of the first flip influence the outcome of the second 
flip? Suppose you flipped a penny 10 times, and each time it came up 
heads. On the 11th flip, what is the probability of getting a head? 
Does the coin remember that it has come up heads 10 times in a row? 
The probability of getting a head on the 11th flip of the coin is 
still 1/2. 

The lottery is the same sort of thing. If 1,2,3,4,5,6 came up on 
Monday, what is the probability that 1,2,3,4,5,6 will come up on 
Thursday? Do you think that 11,12,13,14,15,16 is more likely on 
Thursday than 1,2,3,4,5,6?  Do the little lottery balls remember which 
ones came up on Monday, and thus will let some of the others come up 
on Thursday?

I think both of your questions are the same question. The probability 
of Monday's numbers coming up again on Thursday is the same as for any 
other numbers coming up on Thursday. 

I think your confusion (and it is shared by many) might lie in the 
fact that you are thinking that YOU pick some numbers and then hoping 
THOSE numbers will come up twice in a row. (That is very unlikely.) 
Remember, the first group of numbers is always picked for you in 
Monday's lottery, and you have to match them on Thursday (or they are 
picked for you on Thursday and you have to match them on Monday). In 
either case, you are given numbers that you must match, so what is the 
probability that they will come up in the next drawing? It's still 1 
in 10 million!

-Doctor Bill,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/