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Scratch-off lottery tickets are one of the most widely available forms of gambling that you can take part in. These tickets are usually in the price range of about $1 to $30, and allow you to win prizes that range anywhere from a free chance at another lottery ticket, to jackpots in the tens of millions of dollars.
But what is the true value of a lottery ticket? The state lotteries have to be making a profit, but how large of a profit are they actually making? Are tickets worth 99% of the price you pay for them? 85%? What about 50%? Considering the amount of people that play these scratch off lottery games, it’s kind of surprising how this information is not common knowledge.
In this article, we’ll be going over how to calculate the expected value of a lottery ticket, and finding out how large of a return we should expect to see from our purchase.
First of all, we’re going to need some data. In order to calculate the true value of a lottery ticket, we will need each prize, along with the odds of winning that prize. Fortunately for us, many state lotteries provide these detailed odds on their websites. Today we’ll be looking at the value of scratch offs from the California State Lottery.
The PDF below gives us all of the ongoing games and possible prizes to win, which is all that we need in order to calculate the expected value of a ticket. This chart shows the possible prizes and odds of a $30 game called $10,000,000 Bankroll, which we will be using as our example.
The way that we can calculate the true value of this ticket is by finding the value of each individual prize, and then adding all of those values up.
To find the value of the possibility of winning an individual prize, we can use the following formula:
Let’s first calculate how much the chance to win $10,000,000 is worth. We have a 1 in 3,000,000 chance at winning $10,000,000, so $10,000,000 * (1 / 3,000,000) = $3.33. This means that the chance of winning $10,000,000 from this ticket is worth $3.33 on its own.
When we use this formula for all 12 possible prizes we could win from this ticket and add them all up, we get a total value of $23.96. However, this ticket costs $30 to buy, which means that on average, you will only get about 78.9% of your money back. Compared to other methods of gambling, this is an awful return, considering that games such as blackjack and roulette are upwards of 95%.
But, let’s not look at just one ticket, let’s look at all 57 scratch off lottery games that are currently being advertised. First of all, let’s look at the expected return of the ticket, grouped by the ticket’s price:
We can see that as the price of the ticket increases, the % return increases as well. The return is as low as 59.7% for a $1 ticket, and reaches as high as 80.5% return for a $30 ticket.
So does this mean that we should be buying the more expensive tickets, since they have a higher expected return? Absolutely not. With the more expensive tickets, we are losing a lower percentage of the amount we paid, but we are wagering a larger amount, leading us to a higher total expected loss. The chart below shows the expected loss in dollars grouped by the price of the ticket:
We are only expected to lose about $0.40 when buying a $1 ticket, and we are expected to lose nearly $6 when buying a $30 ticket.
The last metric that we will calculate is the overall probability of winning a prize from a ticket. We can do this by simply adding up all of the odds from our table. For example, if we had a 20% chance at winning $5 and a 10% chance at winning $15, we would have a 30% chance at winning either of those two prizes (20% + 10% = 30%).
After adding up all of the probabilities from the $10,000,000 Bankroll game that we were looking at, the overall probability of winning a prize seems to be 35.9%, which means that you have just over a 1 in 3 chance of winning a prize when buying this ticket.
Let’s take a look at the probability of a winning ticket, grouped by the price of the ticket:
We can see that the more expensive tickets have a higher chance at being a winning ticket, while the cheaper tickets have a worse chance. A $1 ticket has a 22% chance of winning any prize, while a $30 ticket has a 36% chance of winning any prize.
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You will almost never see a scratch off lottery ticket with a positive expected return, but I have seen one occasion where this has happened. If you want to learn more, check out my video on the story of Cash Winfall on YouTube.
Hopefully this article allows you to make more informed choices while gambling on scratch off lottery tickets, and if you’d like to take a look at the odds of winning prizes for any given ticket, you can go to CaLottery’s Website and scroll down and click on “See Odds for all Prize Levels for all Active Scratchers Games”.
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