27 June 2010

Odds of Winning the Lottery

I received a newsletter from a prominent financial and risk services firm where it was stated that the odds of winning the lotto are:

  1. 0,0000072 % with an average pay-out of R2,25 million

  2. The chances of picking five correct balls and one correct bonus ball are only slightly better at 0,000429 %, with an average payout of R173 130,22

  3. Not playing could make you R804 000 better off over your lifetime

This seemed a bit optimistic to me so I investigated the odds of winning the lottery while listening to Cryptopsy and Origin on the Jewtube Relapse Records channel.

How does probability work?

You divide the amount of desired outcomes by the total amount of possible outcomes. If you are casting a dice, you want one number out of six possible numbers. Thus, your odds of getting any one number of a dice are 1/6 or 0,166666667, times 100 to give you 16,6666667 %. For more on probability, see my Bayesian analysis post.

What are the chances of winning the lottery?

This problem could be sub-divided into a few smaller, easier problems using the infamous divide and conquer technique. These sub-problems are:

  1. How many balls are there in total?

  2. How many numbers are there?

  3. How do the combinations work?

How many balls are there in total?

In the South African lottery, or Lotto, there are 49 balls in total. That is, you start off with 49 balls.

How many numbers are there?

The South African lottery has six numbers, plus a bonus ball. Someone who guesses the first six numbers correctly wins over someone who has six numbers correct that includes the bonus ball. For our purposes, I shall exclude the bonus ball because we definitely want the jackpot.

Upon drawing the first ball (or number), there are 49 balls in the pool. You desire one ball out of 49 possible balls (though most guys prefer having two balls, but that's another story). Your odds of drawing this number are 1/49 or 0,020408163.

After the first ball is drawn, there is one less ball in the pool. This means that your odds of drawing the second ball are 1/(49 – 1) , which is 1/48 or 0,020833333.

When drawing the third ball, there are 47 balls left in the pool. This means that your odds of drawing the third ball are 1/47 or 0,021276596.

You get the picture. We can work out the chances of drawing all six balls in a similar way. Since drawing each ball does not depend on drawing a previous ball and we already compensated for having less balls at each round, the event of drawing any ball is discrete. Having discrete balls is awesome because we can multiply each ball event with each other to get the total chances of drawing 6 correct balls. The result is:

1/49 x 1/48 x 1/47 x 1/46 x 1/45 x 1/44 = 1/10068347520

Or one in 10068347520 possible numbers that can be constructed from six numbers each consisting of a number between one and 49.

How do the combinations work?

In order to win the South African Lotto, you do not need to guess the numbers in the correct order. This means any combination of the six numbers could win you the Lotto. In order to work out the number of combinations, we need to work with factorials.

Since there six possible numbers, we need to know how many combinations of six numbers we can possibly have. In other words, in how many different ways may you choose six numbers? This is the factorial of six, or 6!. The answer is 720.

What are the chances of winning the lotto?

The chances or odds of winning the South African lottery or Lotto are one over the total number of possible numbers divided by the total number of combinations of six numbers.

This means your chances of winning the lotto are not even a snowball's hope in hell, but a snowflake's hope in hell. You really want the number? Ok, the answer is 1/13 983 816 or one in roughly fourteen million. Another way to express this is 0,000000072, times 100 to yield 0,0000072 %.

Back to the newsletter

  1. The chances of winning the lottery are not 0,0000072 % but 0,000000072. Update: which of course you have to multiply by 100 to get the percentage, so that's where the 0,0000072 % comes from. Mystery solved!

  2. Using the same method above, the chances of picking five correct numbers out of a possible five are 0,000000524, times 100 to give 0,0000524 %. I'm not sure how the bonus ball would impact this but it seems to me that picking 5 + 1 numbers amounts to picking six correct numbers, since the possible number of numbers and the possible number of combinations remain the same. Thus, provided that I am right, your chances of picking five numbers plus a bonus number are equal to the chances of picking six correct numbers, so that remains 0,0000072 %.

    If I am wrong and the correct answer is 5 correct numbers out of 6, then this becomes the combinations of 6 taken as 5. In other words, 6!/(5!), which is 720/120 or 6. Thus, your chances of picking 5 correct numbers out of a possible 6 are 0,000000429, times 100 to yield 0,0000429 %. Which has one more digit than their number, but it's closer so I guess this is the method that they used.

    I might also be wrong in that 6 combinations plus a bonus ball out of 7 balls were chosen for the original odds. In this case, the answer is the amount of combinations made with 7 numbers, divided by the factorial of 7 taken as 6. The answer is a 0,000000039 or 0,0000039 % chance of winning the Lotto with six correct numbers out of a possible seven, one of which is a bonus.

    This is a bit far from their estimate, so I guess they used the same method I used, except they got a few digits wrong. This might be due to the accuracy of the calculator or spreadsheet application that they used.

  3. The not playing figure assumes that ticket prices remain at R3,50. The playing time is 35 years (between the ages of 25 and 60). Interest is calculated annually at 10 %. This means that not playing Lotto and saving your change, if you played twice a week, yields the handsome sum of roughly R98 991,65 (I am uncertain if this is calculated at simple interest or at compound interest), which is nowhere near R804 000.

    I used a financial calculator, with the values as indicated. I assumed that one ticket is bought for each draw, meaning two draws per week. I multiplied this R7 with the amount of weeks in a year (roughly 52,178571429 weeks) which gave me a yearly investment of R365,25.

    In order to reach their amount of R804 000, you'd have to invest roughly R83 366,52 per year over a period of 35 years at 10 % interest. This amounts to spending roughly R1597,98 per week on the Lotto, which is about R800 per draw or 228 tickets per draw.

Apparently this is a miraculous financial and risk services firm because they can turn R96 552.43 into R804 000 over the same period and using the same interest. Guess I should consider investing with them.

Where do their numbers come from?

Their numbers and my numbers differ by a factor of 100. This is very peculiar, so I found this page on lottery maths that explains the difference.

In the first case, despite their statement that you only play twice per week, they must've bought a total of 100 tickets per lottery draw. This would increase the odds from 1 in nearly 14 million to 100 in nearly 14 million to win a six number lottery, without any bonus. Update: unless of course they were working in percentages, in which case their number is correct.

In the second case, they must've bought a total of 100 tickets per lottery draw. This would increase the odds a thousand times for a five ball plus bonus draw, which gives the ratio between our different numbers.

To put this in perspective, an ordinary calculator cannot even display the minute numbers in question. Regardless of whose method you use, you're likely to end up with 0. In other words, your odds of winning the lotto are as good as 0. The only way to have a noteworthy chance of winning the lotto is to buy a couple of million tickets per draw. But if you could buy a couple of million tickets per draw, you wouldn't need to win the lotto.

Lastly, I multiplied my weekly amount by 100 and worked out the future value of such an investment. This means instead of not spending R365,25 per year on the lottery, you don't spend 100 tickets per week x R3,50 per ticket x weeks in a year per year on the lottery. In other words, you invest this R18 262,5 per year for 35 years at a 10 % interest rate. This investment would be worth about R4 949 582.53 at the end of 35 years. This is again a far cry from R804 000, only this time I am missing about R4 145 582,53.

It appears that this financial and risk services firm does business like FIFA.

1 comment:

Garg the Unzola said...

My brother (who actually studied maths and stats) says there are holes in my lotto probabilities. The following is freely translated from his emails to me:

You have a slight but significant hole in your lotto page: The chances to win the lotto have nothing to do with the amount of combinations. Combinations only come into effect when the order matters - you might as well be working with different shaped rocks here.

Think about it like this: whether you fill in your numbers from lesser than to greater than, or from greater than to less than, or whether you start somewhere in the middle, you still have the same numbers.

In other words, your chances of winning the lotto are something like 1/10 068 347 520 - which is why it is so absurd that someone wins the lotto so often.

FYI, Use it, don't use it..

And the bonus ball scenario is easy. You still have the ordinary 1/(49 * 48 * 47 * 46 * 45) for five balls. Then you must get the sixth ball wrong, in other words you multiply your previous number by 43/44. Then, you have a separate draw where you have to get 1/43, hence you divide your answer by 43.

The order is still irrelevant, as the factorial would cancel on both sides if you insist on using it.

In other words, the odds of getting 5 balls plus the bonus correct are shockingly slim (around 1/10 068 347 520). Compare this with the odds of winning the lotto.. it makes sense mathematically if you only cancel the 43 of the wrong ball with the 43 of the bonus ball. Pretty cool, hey?

Yeah, what he said.

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