Powerball Lottery: Odds of Winning a Prize

Date: 09/02/2001 at 16:16:57
From: Joshua A. Cippel
Subject: Odds of the Powerball Lottery

Dr. Math,

Hello. In the last few weeks, with all of the attention given to it 
in the news media, I have become interested in the Powerball lottery. 
More specifically, the odds of winning. This is not out of any desire 
to win money, but out of simple mathematical curiosity. 

As you may know, the object of the game is to choose 5 numbers from 1 
to 49, and then the "Powerball" number from 1 to 42. After a number is 
drawn to become one of the 5 from 1 to 49, it cannot be drawn again. 
However, any number from 1 to 42 can be drawn as the "Powerball." Of 
course, the person who picks all 6 numbers correctly wins the jackpot, 
but one can also win a prize by picking some of the correct numbers. 
What has me baffled is the statistics of winning. Quoted from 
www.powerball.com, the odds are:  

Match 5+ Powerball = 1:80,089,128
Match 5            = 1:1,953,393
Match 4+ Powerball = 1:364,042
Match 4            = 1:8,879
Match 3+ Powerball = 1:8,466
Match 3            = 1:207
Match 2+ Powerball = 1:605
Match 1+ Powerball = 1:118
Match Powerball    = 1:74

OVERALL ODDS OF WINNING A PRIZE: 1:35

Trying all of the statistics that I know to manipulate these numbers, 
I simply cannot determine how these odds are established. I would be 
very interested if you could help with this puzzle.

Thank you.

Date: 09/02/2001 at 21:33:39
From: Doctor Paul
Subject: Re: Odds of the Powerball Lottery

Let's start with the odds of winning the big prize. To do this, you 
have to pick all five numbers correctly and you also have to pick the 
powerball number correctly.

The answer depends on whether or not order is important and whether or 
not the numbers are "replaced" after they are drawn. Clearly, we are 
working "without replacement" here since a number cannot be drawn 
twice. That is, once a number has been drawn, it is not placed back 
into the pool of eligible numbers.  

In lottery games, the order in which the numbers are chosen is 
unimportant.  All that matters is that you have the same numbers on 
your card as were drawn by the lottery officials.

Since order doesn't matter, we're going to use a combination.  We 
would use a permutation if order mattered.

In general, the number of ways to choose r objects from n when order 
is not important and we are drawing without replacement is given by:

C(n,r) = n!/(r! * (n-r)!)

The probability of winning the big jackpot is given by the total 
number of ways to win divided by the possible number of outcomes. The 
number of possible outcomes (the denominator) is computed as follows:

There are C(49,5) = 49!/(5! * (49-5)!) = 1,906,884 ways to pick your 
five numbers. And there are C(42,1) = 42 ways to pick the powerball 
(you verify the computation using the formula above). Thus there are 
1,906,884 * 42 = 80,089,128 total number of ways that the drawing can 
occur.

To figure out how many ways you can win (the answer is obviously one, 
but we go through the formalities because we'll need them to do the 
next part), think of the balls as being partitioned as follows:

Among the 49 numbers, 5 are "winners" and 44 are "losers." Similarly, 
among the 42 powerball numbers, 1 is a "winner" and 41 are "losers."

If you want to pick all five numbers correctly and pick the powerball 
correctly, then what you want to do is pick 5 out of the 5 winners and 
0 out of the 44 losers on the non-powerball side, and you want to pick 
1 out of 1 winners and 0 out of 41 losers on the powerball side.

This makes the numerator:

C(5,5) * C(44,0) * C(1,1) * C(41,0) = 1 * 1 * 1 * 1 = 1

Hence the probablity is 1/80,089,128

Now let's look at the odds of the "Match 5." Here you have to get all 
five numbers right on the non-powerball side but the number on the 
powerball is not important. Using the tools we developed above, you 
need to pick 5 of the 5 winners on the non-powerball side and 0 of the 
44 losers. But this time, we pick 0 of the 1 winners on the powerball 
side and 1 of the 41 losers.

Hence the numerator is given by:

C(5,5) * C(44,0) * C(1,0) * C(41,1) = 1 * 1 * 1 * 41 = 41

The denominator remains the same for all of these problems. The number 
of possible powerball combinations never changes.

Thus the probability of winning a match 5 is: 41/80,089,128, which is 
pretty close to 1/1,953,393.

For the "Match 4 + Powerball" the numerator would be:

C(5,4) * C(44,1) * C(1,1) * C(41,0) = 5 * 44 * 1 * 1 = 220

Thus the odds here are 220/80,089,128, which is pretty close to 
1/364,042.

You should be able to establish the rest of the formulas pretty easily 
just by following the pattern that has been established. If you have 
any questions, please write back.

The overall odds of winning (1:35) can be verified by adding up all of 
the individual probabilities of winning:

1/74 + 1/118 + 1/605 + ... + 1/80,089,128 = .028706, which is very 
close to 1/35.

I hope this has helped to take some of the mystery out of the odds.  
Again - write back if you need further clarification. If you aren't 
familiar with combinations and permutations in probability, please 
consult a Probability/Statistics textbook or the Dr. Math FAQ:

   http://mathforum.org/dr.math/faq/faq.comb.perm.html   

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

Date: 11/21/2001 at 14:03:32
From: G Saenger
Subject: Question on your powerball odds calcs

I read your analysis on the calculation of the Powerball odds, but I 
have a real problem with the odds of 1:74 that are calculated for 
selecting the correct powerball number only. I realize you have to
take into account selecting 5 incorrect numbers on the non-powerball 
side, and include that calculation, but it still defies logic:

No matter what 5 numbers you select on the non-powerball side (out of 
the field of 49 numbers), you still select one number on the 
powerball side, which has a field of 42 numbers. You are therefore 
selecting 1 number out of a field of 42, and by definition, your odds 
should be 1:42 of selecting the correct powerball number.  It defies 
logic to think you would have worse odds (1:74) than 1:42 in 
selecting the correct powerball number.

Could you please explain this?
Thanks!

Date: 11/21/2001 at 14:40:10
From: Doctor Paul
Subject: Re: Question on your powerball odds calcs

Perhaps it would help to recall what is meant by the probability of an 
event. The probability that an event occurs is given by:

   the total number of ways the event can occur
   --------------------------------------------
      the total number of possible outcomes

In the case above (picking only the powerball number correctly), you
need to count the number of ways to draw only the powerball. This
means you pick the powerball number correctly but it also means that
you don't pick any of the 5 numbers from the non-powerball side
correctly.

As mentioned above, it is helpful to think of the numbers as being
partitioned in the following way:

Among the 49 non powerball numbers, there are 5 "winners" and 44
"losers."

Similarly, among the 42 powerball numbers, there are 1 "winner" and 
41 "losers."

In order to draw only the powerball, you have to pick:

0 of the 5 winners on the non-powerball side
5 of the 44 losers on the non-powerball side
1 of the 1 winners on the powerball side
0 of the 41 losers on the powerball side

In your argument above, you seem to be making the assumption that you
can pick any 5 numbers from the 49 available on the non-powerball side
as long as you pick the powerball number correctly. This is in fact
not the case. What if one of the five you chose on the powerball side
was a winner? Then you wouldn't be in the scenario of picking only the
powerball anymore. You would be in the "Match 1 + Powerball" scenario.

Using the method explained in the link above, the number of ways to
pick:

0 of the 5 winners on the non powerball side
5 of the 44 losers on the non powerball side
1 of the 1 winners on the powerball side
0 of the 41 losers on the powerball side

is:

C(5,0) * C(44,5) * C(1,1) * C(41,0) = 1 * 1086008 * 1 * 1 = 1086008

So the odds of picking only the powerball correctly are:

1086008 / 80089128 = .013559992812

which is very close to

1 / 74 = .013513513514


Now - if we wanted to know the odds of getting the powerball correct
(and we didn't care about what happened on the other side) then we
would just lump all 49 non-powerball numbers together and pick any
five of them. This would give:

C(49,5) * C(1,1) * C(41,0) = 1906884 * 1 * 1 = 1906884

ways of picking our numbers in such a way that guarantees that we get
the powerball number correct.

The probability of this happening is:

1906884 / 80089128 = .02380952381

and this is exactly 1/42.

So the odds of picking the powerball number correctly are in fact 1/42
(as you mentioned should intuitavely be the case). But you have to
realize that the odds of picking only the powerball correctly are in
fact lower because in picking only the powerball correctly, you must
guarantee that you don't pick any of the non-powerball numbers
correctly.

Finally, notice that 1/42 is the sum of the probabilities given above
that are associated with winning the powerball. Here is the chart from
above:

Match 5+ Powerball = 1:80,089,128
Match 5            = 1:1,953,393
Match 4+ Powerball = 1:364,042
Match 4            = 1:8,879
Match 3+ Powerball = 1:8,466
Match 3            = 1:207
Match 2+ Powerball = 1:605
Match 1+ Powerball = 1:118
Match Powerball    = 1:74

If you sum all of the ones that mean correctly picking the powerball,
the answer will be 1/42:

1/74 + 1/118 + 1/605 + 1/8466 + 1/364042 + 1/80089128 = .023761861306

and 1/42 = .02380952381

These numbers are not exactly the same because the probabilities
listed in the chart above are rounded off.

For example - notice that when we computed the "Match Powerball" 
probability above, we got a number that was close to 1/74 but it
wasn't exactly 1/74.

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

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