The Expected Value of Every Major US Lottery Game
We can calculate the 'expected value' of a lottery ticket to determine whether it's a worthwhile investment. Let's delve into the numbers and find out.
The Expected Value of Every Major US Lottery Game
When you buy a lottery ticket, you're not just buying a piece of paper; you're buying a chance, a dream of a life-altering jackpot. But what is that chance actually worth? In the world of mathematics and finance, we can calculate the "expected value" of a lottery ticket to determine whether it's a worthwhile investment. Let's delve into the numbers and find out.
What is Expected Value?
Expected value (EV) is a concept that helps us understand the average outcome of a random event over the long run. It's calculated by multiplying the value of each possible outcome by its probability and then summing all of those values. In the context of a lottery, the expected value of a ticket is the average amount you can expect to win or lose per ticket if you were to play an infinite number of times.
The formula for expected value is:
EV = (Probability of Winning * Prize Amount) - Cost of Ticket*
Powerball: A Deep Dive into the Numbers
Powerball is one of the most popular lottery games in the United States, known for its massive jackpots. A Powerball ticket costs $2. Here's a breakdown of the prize tiers, odds, and the expected value for each prize, assuming a jackpot of $100 million (cash option, which is typically lower than the advertised annuity).
| Match | Prize | Odds of Winning | Expected Value |
|---|---|---|---|
| 5 + Powerball | $100,000,000 | 1 in 292,201,338 | $0.342 |
| 5 | $1,000,000 | 1 in 11,688,053 | $0.086 |
| 4 + Powerball | $50,000 | 1 in 913,129 | $0.055 |
| 4 | $100 | 1 in 36,525 | $0.003 |
| 3 + Powerball | $100 | 1 in 14,494 | $0.007 |
| 3 | $7 | 1 in 580 | $0.012 |
| 2 + Powerball | $7 | 1 in 701 | $0.010 |
| 1 + Powerball | $4 | 1 in 92 | $0.043 |
| Powerball | $4 | 1 in 38 | $0.105 |
Total Expected Value (excluding ticket cost): $0.663
Net Expected Value: $0.663 - $2.00 = -$1.337
This means that for every $2 Powerball ticket you buy, you can expect to lose, on average, about $1.34. Of course, this is just an average. You could win a prize, or you could win nothing at all. But over the long run, the lottery is a losing proposition.
Mega Millions: Is It Any Better?
Mega Millions is the other major multi-state lottery in the US. A Mega Millions ticket also costs $2. Let's look at the expected value for this game, again assuming a $100 million jackpot (cash option).
| Match | Prize | Odds of Winning | Expected Value |
|---|---|---|---|
| 5 + Mega Ball | $100,000,000 | 1 in 302,575,350 | $0.330 |
| 5 | $1,000,000 | 1 in 12,607,306 | $0.079 |
| 4 + Mega Ball | $10,000 | 1 in 931,001 | $0.011 |
| 4 | $500 | 1 in 38,792 | $0.013 |
| 3 + Mega Ball | $200 | 1 in 14,547 | $0.014 |
| 3 | $10 | 1 in 606 | $0.017 |
| 2 + Mega Ball | $10 | 1 in 693 | $0.014 |
| 1 + Mega Ball | $4 | 1 in 89 | $0.045 |
| Mega Ball | $2 | 1 in 37 | $0.054 |
Total Expected Value (excluding ticket cost): $0.577
Net Expected Value: $0.577 - $2.00 = -$1.423
As you can see, the expected value for Mega Millions is even lower than for Powerball. For every $2 ticket, you can expect to lose about $1.42.
Conclusion: A Game of Chance, Not a Financial Strategy
The math is clear: the expected value of a lottery ticket is negative. This means that, on average, you will lose money by playing the lottery. While the dream of winning a life-changing jackpot is enticing, it's important to remember that the lottery is a form of entertainment, not a sound financial investment.
If you enjoy the thrill of playing, by all means, buy a ticket now and then. But don't spend more than you can afford to lose, and don't count on the lottery to solve your financial problems. The odds are stacked against you, and the expected value tells the real story.