Today, I walked by a lottery store in Chinatown and noticed that the Mega Millions prize is now up to $290 million. The Mega Millions was introduced to California in 2005 during Governor Schwarzenegger's tenure. This lottery is a multi-state lottery with steeper odds while offering a bigger purse compared to the original Super Lotto Plus that was introduced in 1986.
While the $290 million is not the largest ever, it is a staggering amount. For historical note, the largest payout from the Mega Millions was $390 million in 2007. The odds of winning the jackpot is currently 1:175,711,536. (I once taught a math class during summer school and asked the students to explain how the odds were calculated. It's actually easier to calculate the odds of a jackpot win than the parimutuel partial wins.)
Many years ago, a friend asked me - since I was a math guy - how to pick the numbers to win. I wasn't a statistician nor a probabilist, so I couldn't offer him a definitive answer. Since then, I've learned a bit more and now have a better answer. First off, the probability of any uncertain event (such as the numbers chosen for this lottery), is based on one's totality of experience. The "true" probability of an uncertain event is not a calculable number.
In the face of the lack of enough experience, one could rely on a calculable number, which I'll call the odds here. In standard parlance, odds and probability are two sides of the same coin. The probability of rolling a 2 on a six-sided die is 1/6 or 1:6 (in odds format) without any additional information. If you know some more information, such as how a player rolls, or that the die is weighted, then the probability will change.
(A very good example of how one's probability can change due to gaining information is the tack flip. Take a standard thumb tack. Not the push-pin type with a long head, but the type with a flat head that when firmly pushed in, is almost flush with the wallboard. Toss the thumb tack in the air and let it land. If it lands with the pin sticking straight up, call that "heads." Call it "tails" if it lands with the pin and an edge of the head touching the ground. Now, the question is, what is the probability of getting heads or tails? Before you go ahead trying to model it and computing a number, assume a probability and then do several tosses. Do several hundred tosses. See what you get and see whether your assessment of the probability of heads (or tails) changed over the course of the trials. Another example would be the spork toss. Take a standard plastic spoon or spork. Toss that in the air and let it land. If it lands with the bowl of the spoon concave up (humped), call that heads. If it lands with the bowl of the spoon concave down (as if can hold water), call that tails. Toss a few hundred times and see whether your assessment on the probability changes.)
Anyway, the answer is not to pick "lucky" numbers, as no one number is any luckier than other. Well, that's not quite true. Don't pick any non-integer, don't pick 0 or any negative integers and don't pick any integer greater than 56 for the first five number or greater than 46 for the power number. Other than that, any allowable number is as good as another.
The answer is when to pick the numbers. The key issue is the expected value of the pay-off. Let's assume you only care about the jackpot (in California, with parimutuel payoff, it's impossible to compute the expected value). The expected value is the prize amount times the probability. The result is a dollar amount that equals the "value" of the ticket. The Mega Millions starts off at $12 million. Thus the expected value is $12,000,000/175,711,536, or slightly less than 7¢. That means, the value of a single pick ticket is 7¢. Of course, you would have to pay $1 to get that.
The value is actually less than the 7¢ when one accounts for the annuity payout and taxes. In many states that play the Mega Millions, you have an option to take the money immediately. In that case, they discount the future value of the annuity to the net present value (NPV). The discounted value is about 60% of the annuity value. So that $12 million, paid out over 26 years, is worth $7.2 million in immediate cash if you wish to take it out as such. Now subtract the federal tax of 30% or so, you net about $5 million. So the expected value of a single-pick winning ticket is 2.8¢. Those are pretty dismal pay-offs. In Vegas and all other state licensed casinos, the expected value for every game of chance is set so that it is no less than 80¢ to the dollar, more or less.
So, when you play is important than what numbers you play, with the lottery. When the jackpot amount equals $175 million or more, then it's time to consider playing. Yes, you have to discount the future pay-out to its net present value, a 40% loss; then another 30% of the NPV for taxes. Basically, take 42% of the jackpot prize amount, and that's your take-home, but since no one ever takes into account this discount and if they managed to win, they're not sorry that it's less than half of the listed prize amount, the listed amount is a good enough gauge.
But there's more. That net amount assumes you're the sole winner. The prize is split equally among all ticket holders, so if there are three tickets sold with the winning number, all three ticket holders take a one-third share of the amount. At the beginning of a rising jackpot, the chances of matched winning tickets is relatively small. But as the jackpot grows, more players enter and the probability of matched winning tickets grow, so the expected value again starts to drop. There's no definitive rule of when the optimal prize amount is, balancing the high pay-off with lack of multiple winning tickets, because the decision on when to play is based on individuals' feelings on whether $100 million is a big amount or whether $200 is a big amount (worthy to put in a buck or two). It may well be that the expected value of the Mega Millions game is never greater than $1.
What's the point of the expected value? Let's illustrate with a simpler model. Suppose I have a four-sided die (tetrahedron die, for those who've played D&D), a perfectly fair die. You pay $1 to play in the game. If I roll a 1, you earn $3 plus your $1 back, for a total of $4. If I roll 2, 3, or 4, you get nothing. The expected value of this game is $1: $4 x .25 (for roll of 1) + $0 x .25 (for roll of 2) + $0 x .25 (for roll of 3) + $0 x .25 (for roll of 4) = $1. If I return $5 (which includes your $1) for a roll of 1, nothing for the rest, the EV is $1.25, and that's is a good bet. If I return $2 (which includes your $1), the EV is 50¢, a bad bet. If the EV is greater than the ante, go for it. If the EV is less than the ante, don't go for it.
But there are other factors in play. There's the person's risk aversion. It's one thing to throw away a buck on a four-sided die game or even the lotto. It's quite another to throw away $100,000 on a four-sided die game with an expected value of $200,000 (return doubles your money). There's been studies (by Tversky and Kahnemann, people I've written about before) that show the irrationality of human decision making under these situations. Another factor in play is the entertainment value of the game. For many people, the 90¢ loss in the difference between the expected value and the ticket cost is considered the "entertainment fee". You're having fun, dreaming of that multimillion dollar home, trips to Ibiza, new Lamborghini. Indeed, poker, a popular game at casino guarantees the house a cut, the vig, so the table, as a whole, loses money to the house. Yet, people have fun playing against each other, enjoying the camaraderie, the drinks, the chatting and so forth. That's the entertainment value of the vig.
Ok, now here's the interesting thing about the Mega Millions. Before 2005, when the Mega Millions entered California's lottery options, the only other million-dollar pay-off game was the SuperLotto. I remember listening on the radio about SuperLotto ad spots telling me of an $80 million jackpot or a $100 million jackpot. I even remember a segment on the Gene Burns show on KGO talking about a huge $200 million or so jackpot. But now that we have the Mega Millions lottery, I rarely see the SuperLotto jackpot over $16 million. Are there more frequent SuperLotto winners? Are there fewer players so that the SuperLotto jackpot doesn't increase as quickly? What's really happening here? When is the SuperLotto ever going to break, say, $50 million? That's the weird curiosity.
While the $290 million is not the largest ever, it is a staggering amount. For historical note, the largest payout from the Mega Millions was $390 million in 2007. The odds of winning the jackpot is currently 1:175,711,536. (I once taught a math class during summer school and asked the students to explain how the odds were calculated. It's actually easier to calculate the odds of a jackpot win than the parimutuel partial wins.)
Many years ago, a friend asked me - since I was a math guy - how to pick the numbers to win. I wasn't a statistician nor a probabilist, so I couldn't offer him a definitive answer. Since then, I've learned a bit more and now have a better answer. First off, the probability of any uncertain event (such as the numbers chosen for this lottery), is based on one's totality of experience. The "true" probability of an uncertain event is not a calculable number.
In the face of the lack of enough experience, one could rely on a calculable number, which I'll call the odds here. In standard parlance, odds and probability are two sides of the same coin. The probability of rolling a 2 on a six-sided die is 1/6 or 1:6 (in odds format) without any additional information. If you know some more information, such as how a player rolls, or that the die is weighted, then the probability will change.
(A very good example of how one's probability can change due to gaining information is the tack flip. Take a standard thumb tack. Not the push-pin type with a long head, but the type with a flat head that when firmly pushed in, is almost flush with the wallboard. Toss the thumb tack in the air and let it land. If it lands with the pin sticking straight up, call that "heads." Call it "tails" if it lands with the pin and an edge of the head touching the ground. Now, the question is, what is the probability of getting heads or tails? Before you go ahead trying to model it and computing a number, assume a probability and then do several tosses. Do several hundred tosses. See what you get and see whether your assessment of the probability of heads (or tails) changed over the course of the trials. Another example would be the spork toss. Take a standard plastic spoon or spork. Toss that in the air and let it land. If it lands with the bowl of the spoon concave up (humped), call that heads. If it lands with the bowl of the spoon concave down (as if can hold water), call that tails. Toss a few hundred times and see whether your assessment on the probability changes.)
Anyway, the answer is not to pick "lucky" numbers, as no one number is any luckier than other. Well, that's not quite true. Don't pick any non-integer, don't pick 0 or any negative integers and don't pick any integer greater than 56 for the first five number or greater than 46 for the power number. Other than that, any allowable number is as good as another.
The answer is when to pick the numbers. The key issue is the expected value of the pay-off. Let's assume you only care about the jackpot (in California, with parimutuel payoff, it's impossible to compute the expected value). The expected value is the prize amount times the probability. The result is a dollar amount that equals the "value" of the ticket. The Mega Millions starts off at $12 million. Thus the expected value is $12,000,000/175,711,536, or slightly less than 7¢. That means, the value of a single pick ticket is 7¢. Of course, you would have to pay $1 to get that.
The value is actually less than the 7¢ when one accounts for the annuity payout and taxes. In many states that play the Mega Millions, you have an option to take the money immediately. In that case, they discount the future value of the annuity to the net present value (NPV). The discounted value is about 60% of the annuity value. So that $12 million, paid out over 26 years, is worth $7.2 million in immediate cash if you wish to take it out as such. Now subtract the federal tax of 30% or so, you net about $5 million. So the expected value of a single-pick winning ticket is 2.8¢. Those are pretty dismal pay-offs. In Vegas and all other state licensed casinos, the expected value for every game of chance is set so that it is no less than 80¢ to the dollar, more or less.
So, when you play is important than what numbers you play, with the lottery. When the jackpot amount equals $175 million or more, then it's time to consider playing. Yes, you have to discount the future pay-out to its net present value, a 40% loss; then another 30% of the NPV for taxes. Basically, take 42% of the jackpot prize amount, and that's your take-home, but since no one ever takes into account this discount and if they managed to win, they're not sorry that it's less than half of the listed prize amount, the listed amount is a good enough gauge.
But there's more. That net amount assumes you're the sole winner. The prize is split equally among all ticket holders, so if there are three tickets sold with the winning number, all three ticket holders take a one-third share of the amount. At the beginning of a rising jackpot, the chances of matched winning tickets is relatively small. But as the jackpot grows, more players enter and the probability of matched winning tickets grow, so the expected value again starts to drop. There's no definitive rule of when the optimal prize amount is, balancing the high pay-off with lack of multiple winning tickets, because the decision on when to play is based on individuals' feelings on whether $100 million is a big amount or whether $200 is a big amount (worthy to put in a buck or two). It may well be that the expected value of the Mega Millions game is never greater than $1.
What's the point of the expected value? Let's illustrate with a simpler model. Suppose I have a four-sided die (tetrahedron die, for those who've played D&D), a perfectly fair die. You pay $1 to play in the game. If I roll a 1, you earn $3 plus your $1 back, for a total of $4. If I roll 2, 3, or 4, you get nothing. The expected value of this game is $1: $4 x .25 (for roll of 1) + $0 x .25 (for roll of 2) + $0 x .25 (for roll of 3) + $0 x .25 (for roll of 4) = $1. If I return $5 (which includes your $1) for a roll of 1, nothing for the rest, the EV is $1.25, and that's is a good bet. If I return $2 (which includes your $1), the EV is 50¢, a bad bet. If the EV is greater than the ante, go for it. If the EV is less than the ante, don't go for it.
But there are other factors in play. There's the person's risk aversion. It's one thing to throw away a buck on a four-sided die game or even the lotto. It's quite another to throw away $100,000 on a four-sided die game with an expected value of $200,000 (return doubles your money). There's been studies (by Tversky and Kahnemann, people I've written about before) that show the irrationality of human decision making under these situations. Another factor in play is the entertainment value of the game. For many people, the 90¢ loss in the difference between the expected value and the ticket cost is considered the "entertainment fee". You're having fun, dreaming of that multimillion dollar home, trips to Ibiza, new Lamborghini. Indeed, poker, a popular game at casino guarantees the house a cut, the vig, so the table, as a whole, loses money to the house. Yet, people have fun playing against each other, enjoying the camaraderie, the drinks, the chatting and so forth. That's the entertainment value of the vig.
Ok, now here's the interesting thing about the Mega Millions. Before 2005, when the Mega Millions entered California's lottery options, the only other million-dollar pay-off game was the SuperLotto. I remember listening on the radio about SuperLotto ad spots telling me of an $80 million jackpot or a $100 million jackpot. I even remember a segment on the Gene Burns show on KGO talking about a huge $200 million or so jackpot. But now that we have the Mega Millions lottery, I rarely see the SuperLotto jackpot over $16 million. Are there more frequent SuperLotto winners? Are there fewer players so that the SuperLotto jackpot doesn't increase as quickly? What's really happening here? When is the SuperLotto ever going to break, say, $50 million? That's the weird curiosity.
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