Lotto Analysis
Inspired by the recent article about people's perception of the lottery as their key to riches, I took it upon myself to compile the following tables to show how you lose money on lotteries.
Here's the rundown for this fictional but representative lottery:
- pick six numbers out of 60
- tickets cost $1
- if you match 3 numbers, you win $3
- if you match 4 numbers, you win $200
Two scenarios for matching 5 or 6 numbers
Scenario 1:
- if you match 5 numbers, the first person to turn in their ticket wins the jackpot/1000
- if you match 6 numbers, the first person to turn in their ticket wins the jackpot
Scenario 2:
- if you match 5 numbers, the first person to turn in their ticket wins the jackpot/500
- if you match 6 numbers, the first person to turn in their ticket wins the jackpot
In reality, if more than one person matches 5 or 6 numbers, they'll just split the money evenly, but to make the analysis easier for me, I'll just assume there is only one winner. By taking a look at the two scenarios, however, it will give me a slightly better picture and allows me to complete my analysis.
The table I generated calculates the expected value for each outcome and adds them up. The expected value is what you would expect on average to occur when playing the lottery. I calculated it by multiplying the probability of the outcome, i.e. matching all 6 numbers, matching all 5 numbers, etc with the payoff for that outcome. I then subtracted the initial cost of the ticket to get the total payoff.
Here is the first table for a jackpot of $10,000,000: matching numbers ways to match probability scenario 1 expected value 6 numbers 1 0.000002% $9,999,999 $0.20 5 numbers 324 0.000647% $9,999 $0.06 4 numbers 21,465 0.042875% $99 $0.04 3 numbers 496,080 0.990894% $2 $0.02 2 numbers 4,743,765 9.475428% ($1) ($0.09) 1 numbers 18,975,060 37.901712% ($1) ($0.38) 0 numbers 25,827,165 51.588441% ($1) ($0.52) Total 50,063,860 100% #N/A ($0.66) matching numbers ways to match probability scenario 2 expected value 6 numbers 1 0.000002% $9,999,999 $0.20 5 numbers 324 0.000647% $19,999 $0.13 4 numbers 21,465 0.042875% $199 $0.09 3 numbers 496,080 0.990894% $2 $0.02 2 numbers 4,743,765 9.475428% ($1) ($0.09) 1 numbers 18,975,060 37.901712% ($1) ($0.38) 0 numbers 25,827,165 51.588441% ($1) ($0.52) Total 50,063,860 100% #N/A ($0.56)
As you can see, the expected value for scenario one is -.66. So for every dollar you play, you would expect to lose .66. Likewise for scenario 2, you lose .56. What does this mean? It means you shouldn't be playing the lotto because you're effectively losing money everytime you play.
Then I asked another question: what jackpot would make this fictional lottery fair? Or in other words, what jackpot would I expect to pay one dollar and get a payoff of $1. Using Excel, I found that the jackpot amount was: $71,258,671.matching numbers ways to match probability scenario 1 expected value 6 numbers 1 0.000002% $71,258,670 $1.42 5 numbers 324 0.000647% $71,258 $0.46 4 numbers 21,465 0.042875% $199 $0.09 3 numbers 496,080 0.990894% $2 $0.02 2 numbers 4,743,765 9.475428% ($1) ($0.09) 1 numbers 18,975,060 37.901712% ($1) ($0.38) 0 numbers 25,827,165 51.588441% ($1) ($0.52) Total 50,063,860 100% #N/A $1.00 matching numbers ways to match probability scenario 2 expected value 6 numbers 1 0.000002% $71,258,670 $1.42 5 numbers 324 0.000647% $142,516 $0.92 4 numbers 21,465 0.042875% $199 $0.09 3 numbers 496,080 0.990894% $2 $0.02 2 numbers 4,743,765 9.475428% ($1) ($0.09) 1 numbers 18,975,060 37.901712% ($1) ($0.38) 0 numbers 25,827,165 51.588441% ($1) ($0.52) Total 50,063,860 100% #N/A $1.46
So you might think, well, if the jackpot gets that high, then it makes sense to play this lottery, and you'd be correct, but only about this fictional lottery. The caveat is it would be WRONG to state the same about lotteries in real-life. That's due to the simplfying assumption we made about only one person winning the jackpot for matching 6 numbers. In the real world, that jackpot would have to be split among the number of people that had the same numbers. And as the jackpot gets higher and more and more people play, it becomes more likely that the number of people with the same winning numbers will be greater than one. As a result, your total payoff decreases and your expected value decreases along with it.
That completes my analysis and is also the reason that I don't play the lottery, and neither should you.
[Update]
An anonymous commentor has made two very good points, I've copied his/her comment below:
Two additional points that make the real-world lotteries an even worse deal:
1. You'd end up paying taxes on the winnings. Depending on your situation, this would probably mean another 30-40% reduction in the after-tax value.
2. The full dollar value is typically paid in an annuity format over 20 or more years. The net present value at the time of winning would be significantly less. Some lotteries allow for a one-time payout, but only at a discount to the "face value" of the winnings.
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8 comments:
Two additional points that make the real-world lotteries an even worse deal:
1. You'd end up paying taxes on the winnings. Depending on your situation, this would probably mean another 30-40% reduction in the after-tax value.
2. The full dollar value is typically paid in an annuity format over 20 or more years. The net present value at the time of winning would be significantly less. Some lotteries allow for a one-time payout, but only at a discount to the "face value" of the winnings.
Tables in Blogger are a pain. I tried creating them in Word and exporting via "Blogger for word" but BFW does not handle most embedded HTML. So your best bet is manual tables...
Oh and now we have mathematical proof of why it's called the "stupid tax"
It's interesting, but I think the bigger thing to consider is that most people who play the lottery don't understand game theory very well, and they probably don't know to call it 'game theory' either. I knew a guy who kept notebooks of lottery results and felt that if XYZ number combination had not come up in 2 years, then its day was due.
I didn't have enough language skill in Korean to tell him that lottery games don't work that way. Each draw is a new game, and it's not a game dependent on the results of the previous game. That's a pretty common myth about playing, i.e. don't play the numbers from the night before. Well, your chances of that combination, or any other combination is the same on the present day.
High schools don't do a very good job of teaching statistics if you ask me. Granted, I took AP Calc instead of statistics my senior year, but I took a lot in college. If more schools taught statistics and its applications, I bet people would understand the lottery and polls a lot better.
It helps if you compare the chances of winning the lottery with other events that might (but are very unlikely) to happen to you in life...like you are 18 to 120 times more likely to die of flesh eating bacteria than to win the lottery. Or my favorite, if you drive 10 miles to purchase your lottery ticket, it's three to twenty times more likely for you to be killed in a car accident along the way than to win the jackpot.
Well yes, you tend to lose when you play the lottery. However, I still think its worth it to play. Fine, call me stupid but consider that all investments have a risk/reward ratio. The higher the risk, the higher the reward. The lottery offers a high risk (ok, *really high*) on a neglible investment and a high reward. As long as I chose to buy a lottery ticket over some other form of consumer spending (lets say my afternoon pack of M&Ms) I am actually acting in a financially responsible fashion choosing "investment" over consumption.
Please see my responses in the following post:
http://franksatheisticramblings.blogspot.com/2006/01/lotto-redux.html
Thanks for all the comments everyone!
But the thing is, the lottery gives all of us a nice daydreaming fantasy. Sure some people are "stupid" and throw away money that they don't have playing the lottery, but lots of other folks play the game for FUN. Just look at Vegas, sure the odds are much better, but people gamble just as much for amusement and the idea of winning as they to for the win itself.
--Mark
www.lottoroller.com
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