Investing can be tough some times. Wouldn’t it be great if you just won the lottery? Man, that would set you for life. Seriously.

Now, we’re not suggesting you run to the closest mini-mart with your life savings. The odds of winning the Big Game (a multi-state lottery open to adults in Massachusetts, Maryland, Georgia, Illinois, Michigan, New Jersey, and Virginia) are roughly 70 Million to 1. That is only slightly better than your chances of making money going long on Overstock.com.

So we all know playing the lottery is a suckers game, but does that mean you should avoid it entirely. The question came up for me when a group at the office I work at started a lottery pool. These are “clubs” where co-workers pool money together in order to increase their chances of winning. If the group (with about 20 members now) hits the $300 Million jackpot, they’ll split it 20 ways. Cut $300M in half for taking the cash option and split it 20 ways, and you get $7.5 Million per person. Not too shabby.

Anyway, I didn’t contribute to the pool the first few times they played since I know how unlikely it is to hit the lottery. Take this analogy from an article by Kevin Devlin entitled Lottery Mania:

Well, imagine laying standard playing cards end to end from New York to San Francisco. The underside of just one of those cards is marked. Start to drive across country, and at some point stop and pick up a card. If you’ve chosen the marked card, you win the jackpot. Chose any other card and you lose. How much would you be willing to pay to play this game? In terms of the odds, you’ve just played the Big Game.

Makes you feel good about blowing your weekend finding a good oil stock to invest in, doesn’t it? Still, even though the odds are so terrible, there is a time to play. I mean isn’t it work about $3 per week to ensure that if my entire office became millionaires I wouldn’t be left out. “Jason, we hit. We hit! Oh… you didn’t enter the pool this week? I’m so sorry. Well, good luck with the project; I’m off to Fiji.” This kind of reasoning aligns with Kevin Devlin’s suggestion for playing the lottery (emphasis mine):

However you look at it, the odds against winning the Big Game jackpot are truly staggering. Does that mean that the best strategy is not to play at all? Oddly enough, the optimal strategy is to

play, but to restrict your wager to an amount of money that is truly of no value to you.

Optimal strategy #1 is to play the lottery with a sum of money “that is truly of no value to you”. $3 a week is right about at my “no-value” level; so I shouldn’t feel bad blowing it on the lottery. In addition, $3 is well worth the entertainment I get from anticipating the results and the comradeship that is created by sharing the moment with my co-workers.

So what is optimal strategy #2? While strategy #1 focuses on minimizing the impact to your finances (in perception at least), strategy #2 will focus on maximizing your odds to win. To understand this strategy, I’ll describe a very simple lottery scenario.

The PA State Lottery Association has gone crazy. They’ve setup a lottery in which the drawing is of just 1 number, ranging from 1 to 3. In any given day, the lotto number could be 1, 2, or 3. Let’s figure the odds if you played $1 (or 1 entry) per day for 3 days. This is easy to do; just multiple the chance of losing in any given day by 3 and subtract that from 1.

2/3 * 2/3 * 2/3 = ~0.3 or a 30% chance of hitting nothing over 3 days.

1.0 – 0.3 = 0.7 or a 70% chance of winning over 3 days.

What if you just played $3 on the first day? Obviously your chance of winning would be 100% since you could cover all 3 numbers. The lesson here is that playing all of your lottery budget in one drawing is going to get you better odds that playing every day or every week.

Now don’t get too excited. The difference in chance of winning using this strategy goes down the larger the possible combinations are. So you’re not going to increase your chances of winning the Big Game by 30%. Let’s do some calculations to see how much you would increase you chances by playing $156 in one drawing rather than $3 per week.

Formula:

1 – [chance of losing] ^ [number of times playing]

$3 per Week:

1 – (69,999,997 / 70,000,000)^52 = 0.00000233%

$156 in One Shot:

1 – (69,999,844 / 70,000,000)^1 = 0.00000233%

What gives here! Well, turns out that my calculator (Excel) can’t even tell the difference in strategies with these huge numbers. But if you’re an astronomer (astrologers have their own strategy by the way) or used to dealing with really big numbers, you might be moved by strategy #2.

Also notice that strategy #2 goes completely against strategy #1. $156 *is* an amount of money that has meaning for me. So it would be hard to part with on a long-shot gamble.

Hopefully we here at InvestorGeeks and others in the blogging community can help you think of some more fruitful things to do with your money. But you also shouldn’t feel bad playing a few bucks each week as long as you won’t miss the money. Good luck.

## Comments (7)

Placing $3 on the PA state lottery doesn’t give you odds of 100% because the first two wagers don’t affect the outcome of the randomly selected number.

It’s actually 70% just like spreading $1 over 3 days.

I don’t get your objection. I assume we’re talking about my hypothetical 3-number lottedy. If there were only 3 numbers, I could buy all possible outcomes for $3. In that situation, I can’t possibly lose.

Do you pick 1, 2 or 3? Or does the computer spit out a number for you like in Big 5 or whatever?

When you play the lottery, you can choose your own numbers. Or you can ask the computer to pick some random numbers for you. However, the machines are smart enough to pick different numbers for each ticket if you ask for more than one. So if I bought 3 tickets for a 3-number lottery, I would get one ticket for 1, one ticket for 2, and one ticket for 3.

If I was getting a random number every time, you are correct that the chances would be the same. But then there is no point in buying more than one ticket with the same number… of course it really depends on how the payout is structured. Is it the same amount no matter how many tickets win (like a daily lottery with 4 digits) or is it one payout that gets split (like the Big 5)? I was just trying to prove the point that your chances are better (although slightly in real life cases) if you buy all your tickets on one day rather than over a period of time. So I avoided these issues.

All of these are really good questions though, and they bring more to mind. Like:

– So obviously you’ll do better in the 3-number lottery if you buy more than 70% of the numbers. What would happen if you bought 2 tickets per day in a 5-number lottery instead of 1 ticket per day? Is that a better strategy? I don’t have time for the math now, but I would guess yes.

– Even in the Big-5 style lotteries, you could buy more than one of the same number. Though it’s not optimal, if you were worried that there would be a split pot, you would get a larger piece of the pie if you had more than one ticket with the winning number. The reason this isn’t an optimal play though is that it is better the spend your money to cover more numbers (doubling your chances to win the whole thing) than buy insurance to win a slightly larger piece of just part of the pot in the case that your number hits.

Okay, didn’t know you could pick your numbers. Interesting… I wonder why PA did that?

Hi I have a problem. Its going to the local gaming facility and spending 100.00 to 200.00 per time and I am tired of losing it all. I have a former friend who plays only 3 of 4 lottery and he hits more often than you would think. At least 8 times per month. He gambles big. The other problem is to find a simple way of playing because Im not good at math. Given example of my knowledge the furthest i can go is fractions or decimals but not fractions into decimals. The state im from is R.I. and they have the lottery 7 days a week at 7pm. I would like to play but dont know how to break it down to a formula. If I can only play 3 instead of 4 the payout is 500.00 per game and if i can hit that 3-4 times a week that would be better than sex. Can you help me figure a simple formula ? Thanks Mark

Mark. I’m sorry to say that I can’t help you figure out a simple formula. While the scheme described in the article above will improve your chances while playing the lottery, it is only by a VERY VERY SMALL amount. And even then, the lottery is still a losing game (meaning in the long run you will always lose at this game).

The main point you should take away from this article is that you should never play the lottery to make money. Remember, the optimal strategy is to play an amount that is negligible to you.

The lottery is one of the worst ways you could gamble. For instance, in a game like Roulette the house’s “edge” is 5.26%. Meaning for every $100 you bet, you lose on average $5.26. The state’s edge in a typical daily lottery is 50%. So for every $100 you play, you lose on average $50. Playing the lottery is about 10x worse than playing Roulette.

Here are some more detailed calculations…

The payout for the 3 number lottery is $500. However, you only have a 1-in-1000 chance of winning. So if you play 1000 times, costing you $1000, you will win on average $500. This is a total loss of $500. Now if your friend is playing a large amount, he may be winning more regularly, but he is for sure losing overall. The same would apply for the 4 number lottery.

I would suggest that you take a serious look at Gambler’s Anonymous. Gambling addictions are very real. Gambling is not bad by itself, but if your gambling is affecting your life in a negative way (either monetarily or by affecting your relationships), you may need help. Below is the link to the GA website. I encourage you to at least take a look.

http://www.gamblersanonymous.org/

Now if you don’t have a gambling problem and want to gamble, but want a chance to actually make money, I would suggest that you move to a game like poker or sports betting. With these games it is possible to have a positive edge (since you are betting against other players rather than a state or casino which is rigging the game to ensure a profit for them). With poker for instance, you could get to a situation where you are making > $1000 for every $1000 you bet (rather than the $500 the lottery is paying out). This is the basis for professional gambling.

http://twoplustwo.com is a great website for reference about poker and other forms of gambling. I would recommend the books by David Sklansky, including Theory of Poker. Take a look at it. If the math seems to complicated, other books by those authors like “Getting Started in Hold ’em” and “Winning Small Stakes Hold’em” by Ed Miller are a great place to start.

On a related note, investing in stocks, either long term or day trading, is a kind of informed gambling. You never know exactly where a stock’s price is going to go, but you can make informed decisions that lead to a positive return.

I’d say good luck now. But I wouldn’t want to encourage you 😉 Let us know how things turn out.