Buying a Lottery Ticket – Is it really worth it?

A common behavioral error occurred this week: Many people thought they could increase their odds of winning the $587.5 million Powerball jackpot by purchasing more than one ticket. On the surface, the logic makes sense. Buy two tickets instead of one and you double your odds. Buy 50 instead of one, and your odds are 50 times better. The problem with such logic is that it doesn’t consider whether buying the extra tickets has any significant impact on the probability of winning.

Powerball, like other forms of gambling, has a fixed number of outcomes. The lottery game picks five unique numbers between 1 and 59. A sixth number, the “Powerball,” is then drawn. The Powerball number ranges between 1 and 35. A total of 175,223,510 combinations can be formed. Since there are a fixed number of combinations, it is easy to calculate the probability of winning the jackpot for any number of tickets purchased. We simply need to divide the number of tickets purchased (assuming each has a different combination of numbers) by 175,223,510.

If you bought one ticket, you had a 0.00000057% chance of winning. Not very good, but a ticketholder in Missouri and a ticketholder in Arizona did win last night. Buying two tickets increased your odds to 0.00000114%. Yes, this was technically twice as good, but your odds were still very low. Splurged and bought 100 tickets (a $200 expenditure)? Your probability of winning only improved to 0.00005707%.

If your goal was to just to have a 1% chance of winning, you would have had to spend $3,504,470, at a price of $2 per ticket. (You would also need several very patient store clerks, the free time to have all of those tickets printed, a system to avoid any duplicate tickets being selected and a method for checking all of those tickets.) Even with the large expenditure, there was a 99% chance you wouldn’t have won the jackpot. You also didn’t have any guarantee of winning enough of the smaller prizes to compensate for the money you spent on tickets.

I bring this up because part of both gambling and investing is understanding the probabilities of winning (making) or losing money. In a game with known fixed odds (e.g., the lottery, poker, roulette, etc.) you can assess how risky a bet is with math. When it comes to investing, the outcome is not always finite (a stock can theoretically keep appreciating in price), but you can still assess the risk by considering what has worked historically. Over the long term, we know value stocks perform better than growth stocks and small-cap stocks beat out large-cap stocks. Companies that initiate or raise their dividends deliver higher total returns than those that don’t pay a dividend. Investment-grade bonds are less likely to default than junk bonds. Funds with lower fees have to beat their benchmarks by a lower margin than funds with high fees to give shareholders the same amount of profit. Diversification and a long-term view will help your portfolio more than a concentration in a few investments and a short-term view.

Keep in mind that, by definition, probability is not the same as certainty. You could do everything right with your portfolio and still not be happy with your returns. Similarly, you could ignore the statistics above, spend $250 on Powerball tickets and win the jackpot. But, if you stop to consider the probabilities of making or losing money, the odds of you making better financial decisions (and not overspending on lottery games) are likely to improve.

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