What is Expected Value?
Expected value (EV) is a concept from probability theory that represents the average outcome of an event over the long run. It is calculated by multiplying the value of each possible outcome by its probability and then summing these values. In the context of the lottery, the EV of a ticket tells you the average amount you can expect to win or lose per ticket if you were to play an infinite number of times.
Calculating the Expected Value of a Lottery Ticket
The formula for EV is:
EV = (P(win) * Prize) - (P(lose) * Cost)
Where:
- P(win) is the probability of winning.
- Prize is the amount you win.
- P(lose) is the probability of losing.
- Cost is the price of the ticket.
For a lottery with multiple prize tiers, the EV is the sum of the EVs for each prize. The EV of a lottery ticket is almost always negative, meaning that on average, you will lose money. This is because the lottery is designed to be a source of revenue for the operator.
When Can the Expected Value Be Positive?
In rare cases, the EV of a lottery ticket can become positive. This typically happens when a jackpot rolls over to an extremely large amount. If the jackpot is large enough, the EV calculation can result in a positive number, suggesting that, mathematically, it is a good time to play. However, there are several caveats:
- Shared Jackpots: The EV calculation usually assumes a single winner. The possibility of sharing a jackpot with other winners reduces the actual prize amount and, therefore, the EV.
- Taxes: Lottery winnings are taxed, which significantly reduces the net prize and the EV.
- Annuity vs. Cash Value: The advertised jackpot amount is the annuity value, paid out over many years. The cash value is a smaller, lump-sum payment that is used for EV calculations.
A Practical Example
Let's consider a simplified lottery with a $1 ticket price. There is a 1 in 10 million chance of winning a $5 million jackpot.
EV = (1/10,000,000 * $5,000,000) - (9,999,999/10,000,000 * $1) EV = $0.50 - $0.9999999 EV = -$0.4999999
In this example, the EV is negative, indicating an expected loss of about 50 cents per ticket.
Conclusion
Expected value is a powerful tool for understanding the financial implications of playing the lottery. While it confirms that the lottery is not a good investment, it can also help identify rare situations where the odds are more favorable. For the vast majority of the time, the EV of a lottery ticket is negative. Smart players understand this and play for entertainment, not for profit.