How Rollover Jackpots Change the Math

How Rollover Jackpots Change the Math

We've previously established that, under normal circumstances, the expected value of a lottery ticket is negative. This means that, on average, you lose money by playing. However, there's a fascinating exception to this rule: the rollover jackpot. When a lottery jackpot isn't won, the prize money is "rolled over" to the next drawing, creating a larger and larger prize pool. This can, in some rare cases, push the expected value of a ticket into positive territory.

The Concept of Positive Expected Value

As a quick refresher, expected value (EV) is calculated by multiplying the value of each possible outcome by its probability and then summing those values. A positive EV means that, on average, you can expect to win more than you spend. In the context of the lottery, a positive EV would suggest that buying a ticket is a rational financial decision.

For a lottery ticket to have a positive EV, the jackpot needs to be incredibly large. How large? It depends on the specific lottery, its prize structure, and the odds of winning. But as a general rule, the jackpot needs to be in the hundreds of millions, or even billions, of dollars.

The Crossover Point

There's a specific jackpot amount, often called the "crossover point," where the expected value of a lottery ticket transitions from negative to positive. This is the point at which the potential winnings from the jackpot are so large that they outweigh the long odds of winning and the cost of the ticket.

Let's consider a simplified example. Imagine a lottery with a 1 in 100 million chance of winning the jackpot. A ticket costs $1. If the jackpot is $50 million, the expected value from the jackpot alone is:

EV (jackpot) = (1 / 100,000,000) * $50,000,000 = $0.50*

If we subtract the cost of the ticket, the net EV is -$0.50. However, if the jackpot rolls over and grows to $150 million, the calculation changes:

EV (jackpot) = (1 / 100,000,000) * $150,000,000 = $1.50*

Now, the net EV is +$0.50. In this scenario, buying a ticket would be a statistically favorable bet.

The Complicating Factor: Splitting the Jackpot

Of course, the real world is a bit more complicated. The biggest factor that complicates the positive EV calculation is the possibility of splitting the jackpot. As jackpots grow, more and more people buy tickets. This increases the likelihood that multiple people will choose the winning numbers, forcing them to share the prize.

When you split the jackpot, your winnings are reduced, which in turn lowers the expected value of your ticket. For example, if you have to split a $150 million jackpot with one other person, your share is only $75 million. This would drop the EV of your ticket back into negative territory.

Calculating the probability of splitting the jackpot is complex, as it depends on the number of tickets sold and the distribution of numbers chosen by players. However, it's safe to say that the larger the jackpot, the higher the probability of a split.

Cash Value vs. Annuity

Another important consideration is the difference between the advertised annuity jackpot and the lump-sum cash value. The advertised jackpot is the amount you would receive if you took the prize as an annuity, paid out over 30 years. The cash value is a smaller, one-time payment. When calculating the expected value, it's more accurate to use the cash value, as this is the amount you would actually receive if you won.

Conclusion: A Rare but Real Phenomenon

While the conditions required for a positive expected value are rare, they are not impossible. When a lottery jackpot reaches a truly astronomical size, it can create a situation where buying a ticket is, from a purely mathematical perspective, a rational decision. However, it's important to remember that even with a positive EV, the odds of winning the jackpot are still incredibly small. You are far more likely to lose your $2 than you are to win hundreds of millions of dollars.

So, the next time you see a massive rollover jackpot, you can impress your friends with your knowledge of expected value. But don't go spending your life savings on lottery tickets. The house always has an edge, even when the jackpot is in the billions.