# Lottery Professional

## The Scenario

You are a professional competition player. You target lotteries under certain mathematical conditions, and make a living from exploiting occasional errors by the organisers of these competitions.

## Math Scenario 1: Buying Every Number

The most recent lottery draw to get your attention has a $20 million dollar prize. Tickets can be bought for $1 each, and there will be 6 numbers selected from 1 to 44. How much money will you need to invest to buy every possible number combination?

To work this out, we need to work out mathematical *combinations* – the order of the numbers drawn doesn’t matter, as long as you have all the numbers. We will need a calculator or computer: we need to work out how many ways we can choose 6 numbers from 44 without replacement (once a number is drawn out it can’t be drawn out again).

The formula for the number of combinations without replacment *c* is:

where *n* is the number of numbers, and *k* is the number drawn.

For our lottery, n is *44* and *k* is 6:

*c = 7059052*

At $1 per ticket, that’s just over $7 million dollars to buy every possible combination.

With a $20 million dollar prize, that sounds like a good deal. The problem is that others will buy tickets too – so you may have to split the prize with others.

For the lottery to be worth targeting, the *average* return on your investment must be $7059052. That means on average you can split the lottery winnings with:

*Average allowable split = 20000000 / 7059052 = 2.833 people*

If we assume that people have bought tickets evenly distributed over the numbers, at what number of total ticket sales (including our 7 million) do we start losing money on average?

*Total allowable ticket sales = 2.833 tickets per combination * #combinations*

*Total allowable ticket sales = 2.833 tickets per combination * 7059052*

*Total allowable ticket sales = 20000000 tickets*

Which of course makes sense – if the tickets are $1 each, then if exactly $20 million dollars of tickets are sold, each ticket holder will *on average* get their money back.

## Math Scenario 2: Set for Life

Many lottery companies give you a choice when you get the money – you can get it all upfront as a lump sum, or as a smaller number of payments over a period of time – what’s known as a “set for life” situation. The advertisements for these lotteries often show a postie delivering the periodic payments to your letterbox.

Let’s say you win one of these, and the choice is between an up front payment of $5 million, or monthly payments of $60000 for 10 years (with the first payment coming at the end of the first month). You can invest the $5 million straight away in a 10 year long term deposit which will provide 5% interest compounding monthly. The periodic payment can be thought of as an annuity with an effective or “nominal” yearly interest rate of 5%. These rates can simply be divided by the number of periods within the year (say 12 in the case of monthly periods) to get the periodic interest rate.

Which choice are you better off with in 10 years time?

### The up front payment:

There are no regular payments here, so we just need to know the value of our initial payment in 10 years time:

*A = P (1 + r/n)^(nt)*

where P is the initial amount, r is the interest rate, n is the number of times the interest is compounded per year and t is the number of years

*A = 5000000 (1 + 0.05/12)^(12 * 10)*

*A = $8235047*

### The periodic payments:

We can use the future value of an annuity formula:

*FV = P * [ (1 + r)^n – 1) / r]*

where P is the periodic payment, r is the effective interest rate *per period*, and n is the number of periods

The effective interest rate per period can be calculated by taking the effective yearly rate and dividing it by the number of periods in a year (12):

*r = 0.05 / 12*

*r = 0.004167*

*FV = 60000 * [ (1 + 0.004167)^(12 * 10) – 1) / 0.004167]*

*FV = $9317136*

So the periodic payments look like a better deal here – in ten years time you’ll be about a million dollars better off.

## Math Scenario 3: House or Gold

Some lottery types allow you to choose between getting a luxury house as a prize or a large amount of gold.

Having won one of these lotteries recently (you’re really good at winning these things), you have to decide whether to take the gold or the house. You don’t intend to keep either – so your decision is going to be based on which one you think you can sell for more money.

The gold prize is 50 kilograms of gold.

The house prize is a 5 bedroom luxury house in the inner city.

Gold prices are currently at $40000 per kilogram. However, you won’t be able to sell at that price – you’ll lose 5% commission in the process. It’ll also take about a week to get the gold and sell it, during which time your accountant estimates prices could vary as much as 3%.

You have the house appraised by 3 real estate agents, who estimate the following sale prices:

$1900000, $2050000, $1840000

The agents will also take a 2% commission on the sale.

If gold prices don’t change and we take the average house estimate, work out which is a better deal:

*Net house profit = average estimated sale price * (1 – commission)*

*Net house profit = (1900000 + 2050000 + 1840000) / 3 * (1 – 0.02)*

*Net house profit = $1891400*

Now for the gold:

*Net gold profit = weight * price / weight * (1 – commission)*

*Net gold profit = 50 * 40000 * (1 – 0.05)*

*Net gold profit = $1900000*

That’s very close – with the gold being slightly more profitable.

Now let’s look at a conservative case, where we assume that the gold price drops 3% and the lowest value house estimate is correct:

*Net house profit = lowest estimated sale price * (1 – commission)*

*Net house profit = 1840000 * (1 – 0.02)*

*Net house profit = $1803200*

Now for the gold:

*Net gold profit = weight * price / weight * (1 – commission)*

*Net gold profit = 50 * 40000 * (1 – .03) * (1 – 0.05)*

*Net gold profit = $1843000*

Once again the gold is a better choice, although by a bigger margin this time.

## Real Life Example – Professional Lottery Syndicates

There have been several examples in real life of syndicates with large amounts of money winning money by targeting certain lottery types.

In 1992 an Australian syndicate bought 5 million out of 7 million possible tickets for a lottery that drew six numbers from 1 to 44, and likely were the sole winner of the $27 million jackpot. They only spent $1 per ticket, so they ended up with a huge profit.

In 2005 a MIT student (it always seems to be MIT students doing this) worked out that a certain lottery’s process meant that tickets could be bought at a profit.