Scratch-Off Expected Value — Full Breakdown & Transcript

[00:00] Playing scratch-off lottery tickets is one of the most widely available ways of gambling, with prizes ranging anywhere from a couple of dollars to the tens of millions of dollars. On the back of these tickets we can normally see the odds of winning a prize,

[00:12] but we don't really know much about the value of the ticket. So in today's video we're going to calculate the expected value of these scratch-off lottery tickets so that we can find the rate of return for these games. A lot of these lotteries will actually have a more detailed breakdown of their prizes on their website, so today we're going to be looking

[00:27] at the California lottery. So for each of the available tickets we have the game name, the ID number of the game, the price of the ticket, the prize level, and the odds of winning that given

[00:39] prize. And this is actually all we need in order to calculate the expected value of each ticket. So as an example we're going to look at this one dollar lucky day lottery ticket. After putting the data into this spreadsheet in this first column here we have each of the individual prizes that

[00:53] we can win and in the next column we have the odds of winning each one of those prizes. Now to find each prizes individual contribution to the overall value of the ticket, we're first going to take the prize, which in this case is $2, and then we're going to take the odds of

[01:07] winning that prize, which is 1 over 17. And when we multiply these together, we see that we get about 12 cents which means that just the chance of winning from this ticket is worth about 12 cents on its own We can now calculate the expected value contribution of all the other prizes

[01:23] and we can sum them all up, and we see that the sum is about 51 and a half cents. So this isn't the final expected value of our ticket because we still have this one in eight chance of winning the ticket itself. So we can take the sum of all the prizes, and we can multiply by a one in eight

[01:40] chance of winning that ticket and we can see that the chance of winning a ticket is worth about six and a half cents. So now we're going to want to add the expected value of winning these cash prizes to the expected value of winning the ticket and we can see that this turns out to be about 58 cents

[01:55] but this is a problem because we just used the 51 and a half cents as the expected value of our ticket in our equation and now this is 58 cents so we're going to have to update our equation. We could continue running through this process until the expected value of the ticket

[02:10] eventually converges, but there's actually a shortcut that we can use. So we're calculating the expected value of the ticket by taking the sum of the expected values of the cash prizes and then we're multiplying the sum by the 1

[02:22] in 8 chance that we win the ticket itself. But now we need to multiply this amount that we added by 1 8th again so now we get 1 8th squared And then this amount that we add we need to multiply by the 1 8th chance that we win a ticket So we get 1 8th to the third and so on until the price of the ticket converges

[02:40] So this is just another form that we can put our equation in. And this is actually a form of the geometric series. And the geometric series has this formula shown above. So we can find this infinite sum and it turns out to be about 59 cents, which is the expected value of our lottery ticket.

[02:56] This number seems surprisingly low, considering other casino games such as slots and roulette pay out about 95% of the bet, which means you're only losing about 5% of what you bet instead of over 40% in this case. But let's calculate the

[03:11] expected value of all these other games first before we start making assumptions based off one scratcher. I found this Python library called Tabula, which is able to extract tables from PDFs, so here we have a table of all of the possible prizes from all of our games. So first I found the probability of winning any prize for each game

[03:28] and then I grouped them by the price of the ticket. So we can see down here for the one and two dollar tickets our chance of winning is about a little less than 25 percent and as the price of the ticket increases the chance of winning actually increases as well and once we get up to the 30

[03:42] dollar tickets we have about a 36 percent chance of winning Next I calculated the expected value for each of the tickets as we did in the spreadsheet and then I plotted the return of the tickets grouped by the price of each game So we can see down here in the to range you getting about a 60 return on each game So a game for example you

[04:00] getting about $1.80 back on average. But as we go into these more expensive games from $5 to $10 and eventually $30, we get up into the 80% return range. So you're expected to get about $24 back

[04:14] on a $30 ticket. Now this doesn't necessarily mean that the more expensive games are better, and we can see that in this plot here. We have the expected loss by the price of the game, and we can see that for the $30 ticket, we're losing about $5 to $6 per game on average.

[04:28] But for the $1 ticket, we're only losing about 40 cents on average. So you actually lose a larger percentage on these lower priced games, but because you're spending so much money on the more expensive games, you actually end up with a higher expected loss as you pay more money.

[04:41] So for any lottery game you find today, the expected value of the ticket is almost guaranteed to be negative. But I've actually seen one example in particular where the expected value of the lottery ticket was actually positive. And that story is actually going to be the topic of my next video. So be sure to subscribe if you haven't already.

[04:57] I'm still in the very early stages of growing this channel, so any support you could show by liking, commenting, or subscribing would be very helpful. So thanks for watching and I'll see you in the next video.