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Powerball Odds When Buying More Tickets

Date: 12/31/2001 at 13:29:27
From: Daren Henning
Subject: Powerball odds when buying more tickets

I've read your articles about Powerball odds but a friend of mine and 
I are having an argument about odds when buying multiple tickets. If 
the odds are 80,000,000 to 1 of buying a winning ticket, he says they 
got to 40,000,000 to 1 if you buy 2 tickets. He goes on to say that if 
you bought 10 tickets it would drop to 8,000,000 to 1 and if 1,000 
tickets are bought it further drops to 80,000 to 1.  

It seems to me that these numbers couldn't drop that drastically from 
such a few tickets being purchased, but I'm not sure how to formulate 
a response to this. My original thought is that if you bought 10 
tickets you would have 10 chances, all with an equal chance of 
80,000,000 to 1 of winning and, that the odds wouldn't be reduced as a 
fraction. Now, maybe I'm thinking you would have a 79,999,990 to 1 
chance of winning if you bought 10 tickets. Please help, since this 
has been an on going argument for some time and it has caused some 
heated, but friendly, debate!

Thank you,
Daren D Henning

Date: 12/31/2001 at 14:05:49
From: Doctor Paul
Subject: Re: Powerball odds when buying more tickets

Your friend is right. Quoting from the Dr. Math FAQ on Probability:   

The probability of winning this second lottery is 1 in 14 million. 
[They're not talking about the Powerball Game here]  What would happen 
if you bought 7 million tickets? 

If you picked a different combination of six numbers for each of those 
7 million tickets, you'd have 7 million of the possible winning 
combinations and the numerator of your probability fraction would 
therefore be 7 million. Given the second lottery, with a sample space 
of 14 million possible combinations, the probability of winning the 
lottery is 7 million/14 million, a probability of 50%. 

Thus you can see that the more tickets you buy, the better your 
chances of winning the lottery. However, you need to buy lots and lots 
of tickets before the number of tickets you hold really makes a 
difference. Even if you buy 100 tickets (which might cost you $100), 
your chances of winning would still only be 100/14 million - not even 
close to a 1% chance.

The key idea here is that your odds of winning increase dramatically 
when you buy more than one ticket. But the odds are still so stacked 
against you that it really doesn't make much of a difference. You may 
be 10 times more likely to win if you buy 10 tickets, but one in eight 
million isn't very good odds either. 

Here's a tip: if you're spending all of your money on lottery tickets 
in the hope of increasing your odds of winning, stop wasting your 
money. :-)

I hope this helps. Please write back if you'd like to talk about this 
some more.

- Doctor Paul, The Math Forum   

Date: 12/31/2001 at 17:14:02
From: Daren Henning
Subject: Powerball odds when buying more tickets

Thank you for your quick response! Another question I have is about 
the odds themselves. In a 50/5 pick with a 36 number powerball, the 
odds are 76,275,360:1. Is this the total number of possible 
combinations for the outcome of the game? The reason I am asking is 
that the question in you archives asks about a lotto with 7,000,000:1 
odds with a 49 and 6 pick. You said that there are something like 
13,000,000 differant combinations here. Was this just a hypothetical 
lotto or would there really be those odds with that many combinations?

Daren D Henning

Date: 01/01/2002 at 03:18:31
From: Doctor Jeremiah
Subject: Re: Powerball odds when buying more tickets

Hi Darren,

The cause of the huge difference in the odds is the extra powerball.  
In a 50/5 lottery the 5 balls can come out in any order, but a 50/5 
lottery with an extra powerball is NOT the same as a 50/6 lottery.

In a 50/5 lottery with no powerball (just the 5 balls), since they can 
come out in any order the formula is:

  ---------- = 2118760:! or about 2 million to 1
  (50-5)! 5!

The powerball is like a separate lottery with one ball. You have to 
win both lotteries and the powerball has odds of 36:1, so you need to 
multiply the odds:

  50/5 + powerball(36) = (2118760*36):1 = 76,275,360:1

A 49/6 lottery with no powerball just has 6 balls, and the odds are:

  ---------- = 13983816:! or about 14 million to 1
  (49-6)! 6!

But if it had a powerball it would be MUCH worse odds.

If I remember correctly the lottery we had where I grew up was a 49/5 
lottery with a powerball drawn from a separate 49-ball bin.

  ---------- = 1906884:! or about 2 million to 1
  (49-5)! 5!

  49/5 + powerball(49) = (1906884*49):1 = 93,437,316:1

Which is only slightly worse than the 76 million to 1 odds of your 
lottery. So the number of combinations is insignificant compared to 
the powerball. It's the powerball that makes the lottery hard to win.

- Doctor Jeremiah, The Math Forum   

Date: 01/01/2002 at 13:20:12
From: Daren Henning
Subject: Powerball odds when buying more tickets

Thank you very much!  This has answered a lot of questions.

Daren D Henning
Associated Topics:
High School Probability

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