Lotto - Positive EV?

Alright so I'm traditionally very good/conservative with my money. I play low HA games in basically the best way possible. I have to admit though, my weakness is throwing a few bucks on the lotto when the jackpot gets up there (Powerball/Mega Millions 100+ million).

Currently Mega Millions is at $290M (~$183M instant payout). I consider the instant payout to be the most accurate since all annuities do is lower the current value of your future payment. For this problem let's ignore taxes (although I believe taxes need to be considered because they're equivalent to a vig in the player's eyes).

Mega Millions is played with a field of 5 numbers ranging from 1-56 that cannot be repeated and their order does not matter. In addition, you must also hit the Mega Ball number that ranges from 1-46. I have a couple of questions:

1. How do you calculate the odds of hitting the jackpot ONLY? Please go into details, I'm getting something way off.

2. The Washington (and Mega Millions) websites say that there is a 1:175,711,536 chance of hitting the jackpot. To reference the previous question, I get well over 400 million to 1 when just counting the 5 numbers from 1-56. If it is 1:175,711,536 - does that mean it would be advantageous for some millionaire to buy one ticket of every combination if he knew nobody else would hit the jackpot (This could lead to a -real- discussion since the chance of someone else hitting the jackpot is something you need to consider)? If it is not 1:175,711,536 - how are they getting that number?

Reference for advertised odds: Mega Millions Official Website

Quote: ahiromu

...1:175,711,536 ...

46 * 56*55*54*53*52/(5*4*3*2*1) = 175,711,536

Quote:

... I get well over 400 million to 1 when just counting the 5 numbers from 1-56...

There are 56*55*54*53*52 = 458,377,920 permutations for picking five numbers from 1-56. However, the order of the five numbers does not matter, so 458,377,920 must be divided by 5*4*3*2*1 to find the number of combinations for picking 5 numbers from 1-56.

Here is a discussion of having to share a lottery jackpot with other winners: http://wizardofvegas.com/forum/off-topic/off-topic/2447-split-lottery-jackpot-winners/

Oh I see, when you do the initial multiplying of the 50's you need to divide it. Thank you, I was having a problem dealing with the Mega Ball number.

My rule of thumb is the lottery is +EV when the jackpot is 4x the odds. If the odds to win Mega Millions is 1 in 175M, you need to be looking at a pot of around 700M in order to play profitably on the jackpot (I'm ignoring the other pays). Why? First off, you always want to take the lump sum payout, but that's only about 1/2 the pot. They just use it to buy a 20-year annuity to double the money, but doubling your money in 20 years is a suboptimal investment - you can do better on your own. Second, you'll lose about half of what's left to taxes. And all that assumes you won't split the pot with anyone, which is more likely if you pick numbers below 32. So pick numbers above 31 when the pot gets big and good luck.

As far as cornering the market goes, yes, if you buy one of every ticket combination, then you'd be spending $175,711,356, and you would need to win at least $175,711,357 to have a positive EV. Since you would win $31,979,120 from the smaller prizes, you would need to win at least $143,732,417 from the jackpot to be +EV. This would need to be after taxes, splits, tips on the hand pay, etc., and doesn't take the Megaplier into account at all (but the jackpot would need to be much larger, even at ×4).

Of course, this presumes that you can generate 700 tickets per second. ;^)

There is a history of at least one team doing just what you proposed. They had 175 million pencil marked tickets waiting to go, divided amoung the team. When it got big enough, they hit a bunch of lottery agents - who may have been in on the deal and kept their stores open.

The biggest monkey wrench to the whole problem, is the possibility of sharing the prize with multiple winners. I believe the team waited for the prize to be 2.5 times the cost.

FYI: Taxes aren't as big of an issue as you may think. Lottery losses get deducted right off the top. Since you'll have all those losing tickets, only the profit is taxed.

Quote: MathExtremist

... First off, you always want to take the lump sum payout, but that's only about 1/2 the pot. They just use it to buy a 20-year annuity to double the money, but doubling your money in 20 years is a suboptimal investment - you can do better on your own. ....

ME, I am going to make the assumption that you did not make the mistake of looking at the lump sum amount, calculating how much you could turn it into after twenty years (actually 25 for Mega-Millions and 29 for Powerball), and comparing it to the advertised payout amount. That would be an inappropriate comparison, because the advertised figure is the annuity amount (payouts all along the way) and not an end-of-investment figure. Many people make that mistake in doing the comparison, but I doubt that you would.

However, have you taken into account the tax deferral aspect of the annuity? I think that is where a jackpot winner draws the big benefit of not taking the lump sum. By letting the lottery corporation make the investment, you allow them to invest pre-tax dollars, which is a much larger sum than you can invest yourself with the lump-sum, after-tax amount. To do better, you would need to obtain substantially higher returns than the annuity rate.

Also, as for "you can do better on your own", are you referring to risk-free investments, or do you discount your expected returns based on the risk to which you expose the principal? The lottery annuity funds not only are substantially invested in federal government securities, but the jackpot winner also has a binding contract with the state to pay the annuity proceeds even if the lottery corporation's investment does not pay out as expected. I think that's about as risk-free an approach as is available these days.

If I were actually to win one of those mega-jackpots, I would have an additional, non-mathematical reason for preferring the annuity. Considering the level of wealth I have personal experience managing, I would be well over my head trying to manage hundreds of millions of dollars. I think it would be a tremendous hassle to do that, and if I had that kind of personal wealth, the last thing I would want is added hassle in my life. Yes, of course I could hire a professional financial adviser, perhaps someone who would turn out to be (ethically) a second cousin of Bernie Madoff. Personally, I'd prefer the ease of just having the checks come in on a very-reliable, annual basis while I enjoyed my hassle-free life.

I can see the $175,000,000 in cash being a small problem :)

Quote: Doc

If I were actually to win one of those mega-jackpots, I would have an additional, non-mathematical reason for preferring the annuity. Considering the level of wealth I have personal experience managing, I would be well over my head trying to manage hundreds of millions of dollars. I think it would be a tremendous hassle to do that, and if I had that kind of personal wealth, the last thing I would want is added hassle in my life. Yes, of course I could hire a professional financial adviser, perhaps someone who would turn out to be (ethically) a second cousin of Bernie Madoff. Personally, I'd prefer the ease of just having the checks come in on a very-reliable, annual basis while I enjoyed my hassle-free life.

I would imagine at $290 million even using the annuity would require a financial planner of some sort.

Your other points are totally valid, though. Why do most people seem to take the lump sum, then? Greed? They don't trust the lottery?

Quote: ElectricDreams

Why do most people seem to take the lump sum, then? Greed? They don't trust the lottery?

Probably because the state encourages it.

I have a buddy that works for the NJ Lottery. Here, (and presumably elsewhere), if you win, and have selected the annuity, you are given the chance to change your mind. If you selected cash when purchasing the ticket, you can not change. They have trained the ticket agents to automatically hit the cash button rather than ask, when a customer doesn't specify a choice.

Why?

Because, as shocking as it is, people with million dollar annuities sometimes move without leaving a forwarding address. The headaches it causes the Lottery Commission are so great that they prefer people take the cash option.