You’ve often heard that it is an incredibly dumb idea to buy a lottery ticket — that you’re basically just wasting your money. Yet, a lot of people buy lottery tickets anyway, and will continue to do so. Why is this? How much money can we expect to win off a lottery ticket?
Framing the Question
First it should be obvious to anyone that buying a lottery ticket is going to be a losing proposition. If we could expect to get a lot of winnings off a lottery ticket, then everyone would buy lottery tickets, and lotteries would go out of business. The lottery has to win off the average ticket, or there would be no incentive to run the lottery.
However, perhaps we can ask this question differently. We know that we will end up losing money if we buy a lottery ticket. But how much money do we expect to lose, per ticket, on average?
So what is expected utility and how does it work? Well, this is basically how much money we would expect to receive / “win”, on average. If there was a ticket that we paid $3 for and won $5 all the time, this would be an expected utility of $2, because we would always win $5 and pay $3, netting $2.
But how does this work when there is only a probability of winning? To do this calculation, we multiply the winnings by the probability of winning. Why? If a ticket costs $3 and has a 50% chance of winning $10 (and a 50% chance of not getting anything), we have the following as all possible options:
- 50% chance of paying $3 and winning $10, netting $7. (Utility is $7)
- 50% chance of paying $3 and winning $0, netting -$3. (Utility is -$3)
If we were to buy 10 tickets, we would expect 5 of them to get us $7 each and 5 of them to lose us $3 each, because that’s how probability works. Adding up the totals, buying 10 tickets should net us $7*5 + -$3*5, or $20.
We can get this via a formula: (Tickets Bought)*(Chance of winning)*(Winnings) + (Tickets Bought)*(Chance of losing)*(Losings)
For our above example, we do (10)*(0.5)*(10-3) + (10)*(0.5)*(0-3) = 20.
In order to determine if we should buy this ticket, we preform this calculation on one ticket, or (1)*(0.5)*(10-3) + (1)*(0.5)*(0-3). It equals $2 winnings on average, so we should buy this lottery ticket. (Knowing that this is on average is important, because no ticket actually wins us $2.)
Multiple Ways to Win?
However, what if there were multiple ways to win? For example, what if we had the following hypothetical ticket that costs $10 to buy:
- 1% chance of $100,000 – (Utility = $99,990)
- 2% chance of $10,000 – (Utility = $9,990)
- 3% chance of $1,000 – (Utility = $990)
- 4% chance of $100 – (Utility = $90)
- 5% chance of $10 – (Utility = $0)
- 85% chance of winning nothing – (Utility = -$10)
Well let’s simulate this by buying 100 tickets with the expectation of even probability: 1 ticket will get us $99,990; 2 tickets will get $9,990 each; 3 tickets will get us $990 each; 4 tickets will get $90 each; 5 tickets will net us nothing; and 85 tickets will lose us $10 each.
If we total it up, the 100 tickets should win us $122,450. Clearly we’ve hit the jackpot, we definitely should buy this ticket a lot! From this, we can expect a single ticket to net us $1224.50 on average! They won’t be in business for long.
Making a Formula
So from this, we now can get a new formula that takes into account all the paths to various winnings:
[T * P1 * (W1 - C)] + [T * P2 * (W2 - C)] + [T * P3 * (W3 - C)] + … + [T * Pn * (Wn - C)]
This formula is for a ticket with n ways to win. T is the number of tickets we buy, Pn is the probability of winning using the nth way, Wn is the amount of winnings received using the nth way, and C is the cost of the ticket.
Plugging in our values from the hypothetical ticket, we get:
[1*0.01*(100000-10)] + [1*0.02*(10000-10)] + [1*0.03*(1000-10)] + [1*0.04*(100-10)] + [1*0.05*(10-10)] + [1*0.85*(0-10)]
[999.9] + [199.8] + [29.7] + [3.6] +  + [-8.5]
Which brings us to the total we figured out without the formula, a net winnings of $1224.50 per ticket.
Applying This to A Lottery
So now let’s look at an actual lottery ticket, from MegaMillions. In this standard lottery, you pick five numbers, each number between 1 and 56, and then a sixth number between 1 and 47 that is the Mega Ball.
The MegaMillions site gives the lottery odds, so we know the winnings pay out like this:
- 0.000000569% chance of Jackpot, currently $40 million
- 0.0000256% chance of $250,000
- 0.000145% chance of $10,000
- 0.00653% chance of $150
- 0.118% chance of $10
- 0.327% chance of $7
- 0.709% chance of $3
- 1.333% chance of $2
- 97.5% chance of $0
(Note: because some tickets overlap, the distribution is not perfect. I took out the second way to win $150 to help correct for this.)
So now we can plug this into our formula for buying 1 ticket.
[(1/175711536)(40000000-1)] + [(1/3904701)(250000-1)] + [(1/689065)(10000-1)] + [(1/15313)(150-1)] + [(1/844)(10-1)] + [(1/306)(7-1)] + [(1/141)(3-1)] + [(1/75)(2-1)] + [(39/40)(0-1)]
Evaluating that, I get:
[0.22765] + [0.06403] + [0.00145] + [0.00973] + [0.01066] + [0.01961] + [0.01418] + [0.01333] + [-0.975]
Which gives us a net winnings of -$0.61 per ticket, on average. This means that we should never buy this lottery ticket, because we would expect to lose money with each purchase. If you played the lottery for a decade, once per week, your losses would end up totaling $318.
Making the Jackpot Worth It
So now that we know it’s probably a bad idea to play the lottery, how do we “fix” it? Of course, this fix would never happen in real life since it would destroy the massive profit lotteries expect to receive, but it’s still an interesting application of our formula. First, how large would the jackpot have to be in order to make buying the ticket worth it, assuming the odds of winning and other payouts don’t change?
To solve this, we apply our forumla in a different way. Specifically, we want it to equal… say $1.01, so we know that we win a penny off of every ticket. Let’s use algebra. J is the Jackpot:
[(1/175711536)(J-1)] + [(1/3904701)(250000-1)] + [(1/689065)(10000-1)] + [(1/15313)(150-1)] + [(1/844)(10-1)] + [(1/306)(7-1)] + [(1/141)(3-1)] + [(1/75)(2-1)] + [(39/40)(0-1)] = 1.01
Narrowing that down with the calculations from last time, we can solve for the J that makes buying lottery tickets worth it.
[(1/175711536)(J-1)] + [0.06403] + [0.00145] + [0.00973] + [0.01066] + [0.01961] + [0.01418] + [0.01333] + [-0.975] = 1.01
[(1/175711536)(J-1)] – 0.84204 = 1.01
[(1/175711536)(J-1)] = 1.85204
J-1 = 325424793.13344
J = $325,424,794.13
So until the jackpot increases to over $325 million, it’s not worth playing the MegaMillions at the current odds.
Making the Ticket Price Worth It
We can now do another interesting application of our formula to make a “fix”. What price would the ticket have to be to make buying the lottery ticket worth it, assuming the odds and payouts don’t change?
Let’s algebra again, using P for price:
[(1/175711536)(40000000-P)] + [(1/3904701)(250000-P)] + [(1/689065)(10000-P)] + [(1/15313)(150-P)] + [(1/844)(10-P)] + [(1/306)(7-P)] + [(1/141)(3-P)] + [(1/75)(2-P)] + [(39/40)(0-P)] = P + .01
(40000000/175711536) – (P/175711536) + (250000/3904701) – (P/3904701) + (10000/689065) – (P/689065) + (150/15313) -(P/15313) + (10/844) – (P/844) + (7/306) – (P/306) + (3/141) – (P/141) + (2/75) – (P/75) – (39P/40) = P + .01
(40000000/175711536) + (250000/3904701) + (10000/689065) + (150/15313) + (10/844) + (7/306) + (3/141) + (2/75) – (P/3904701) – (P/175711536) – (P/689065) – (P/15313) – (P/844) – (P/306) – (P/141) – (P/75) – (39P/40) – .01 = P
(40000000/175711536) + (250000/3904701) + (10000/689065) + (150/15313) + (10/844) + (7/306) + (3/141) + (2/75) – .01 = P + (P/3904701) + (P/175711536) + (P/689065) + (P/15313) + (P/844) + (P/306) + (P/141) + (P/75) + (39P/40)
0.38864667 = (104313385245792122043967386751 * P) / 52158117663020167059094033200
(0.38864667 * 52158117663020167059094033200) / 104313385245792122043967386751 = P
P = 0.19432
So if the ticket was repriced to be 19 cents instead of a dollar, it would be worth buying.
If your motivation is to make more money than you spend, playing the lottery is a terrible idea. However, the lottery can be a great way to see probability in action.
Also, it’s a great way to see how common misconceptions of probability work to people’s disadvantages. Buying more tickets does not increase your chance of netting money, because your cost goes up along with the amount of tickets you buy as well.
Furthermore, the fact that you lost on your previous ticket does not mean that your chance of winning the next ticket is higher — the actual percent changes of winning do not go up or down (at least not noticeably to the point where it would make a difference).
Lastly, as stated on this website about reasons people play the lottery:
Ironically, the low odds of actually winning the lottery are overlooked by most players or they simply do not care. Research has proven that when humans are convinced a low probability event can occur they will overestimate the chances of winning despite the fact that the majority of players will never win.
This website also puts in perspective, the only legitimate reason to play the lottery is that you have fun… losing money.
Also, if you think the lottery exists to sell hope, you ought to request the New Improved Lottery.
I now blog at EverydayUtilitarian.com. I hope you'll join me at my new blog! This page has been left as an archive.