Powerball demystified

The US Powerball lottery hysteria took another step when no one won the big jackpot in the last draw that took place on October 20, 2018. So, the total jackpot is now 2.22 billion dollars. I am sure that you want to win this jackpot. I myself want to win it.

Actually, there are two different lotteries: The Mega Million lottery prize is about 1.6 billion dollars, and the probability of winning when playing a single ticket is about 1 in 302 million. The Powerball lottery jackpot is “only” 620 million dollars but the probability of winning is slightly better: about 1 in 292 million.

The probability of winning both jackpots is therefore the multiplication of the two probabilities stated above, which 1 about 1 in 88000000000000000.

Let’s not be greedy, and aim just to 1.6 billion jackpot, although its probability of winning is slightly worse.

First, it should be noted that although the probability of winning is small, it is still positive. So if you buy a ticket you get a chance. If you do not buy a ticket you will not win, period.

Second, is buying a ticket a good investment? It looks like it is. The price of a ticket is two dollars. On the average, you will win the jackpot with probability of 1 to 302 million, and lose your two dollars dollar with probability of nearly 1. Therefore, your average return is about the jackpot multiplied by the probability of winning it minus the price of the ticket. Since the probability of winning is 1 in 302 million and the jackpot is 1600 million, then the expected return is 1600/302–2 , which is positive — about 3.30 dollars. Therefore, you should play. Or shouldn’t you?

The above figure — expected value of 3.30 dollars is an expectation of money. It is not money. You are not going to gain this expected sum of money when you play the lottery once. You either win the jackpot or lose your money. Of course, if you get a chance to participate in such a lottery with such a jackpot as many times as you wish, you should play, and the law of large numbers will be in your favor. This is not going to happen, of course. You only get to play this game once.

The next interesting question is what is the probability that someone will win?

Assume that you roll a die. The probability of rolling 6 is 1 to six. If two people roll a die, then the probability of at least one of them rolling six is about 1 in 3.3. If 3 people roll a die then the probability of at least one of them rolling six is even better: 1 in 137, and so on. The lottery is similar. Think of a lottery ticket as a die, only that the probability of rolling 6 is 1 to 302 million. If two people are rolling suck a dice, i.e. buying a lottery ticket, then the probability that at least one of them rolling a six is slightly better than 1 to 302 million. How many people should buy a lottery ticket to make the probability of a least one win greater than 5%? 10%? 50%? What is the probability that two or more people will share the jackpot? These probabilities depend on the amount of tickets sold. The more ticket sold, the higher the probability that someone wins. If you know the number of tickets sold, you can be approximated these probabilities using the Poisson distribution. You can also back-calculated the number of tickets need to be sold in order to set the probability that someone wins to any level you like. I’ll skip the technical details. According to my calculations, the number of tickets need to be sold to ensure that the probability of at least one winner exceeds 0.5 is about 210 million.

But wait: the price of a ticket is 2 dollars dollar, and there are only 302 million possible combinations of numbers. So, if I buy all possible tickets, it will cost me only 604 million, and I am guaranteed to win 1.6 billion. This is a net profit of nearly a billion dollars. Not bad. Can I do it?

The short answer is “Yes”. The long answer is “probably not”.

Yes. It was done before. In 1992, the jackpot of the Virginia lottery was 27 million dollars, while the probability of winning the jackpot was about 1 to 7 million, and the price of a ticket was 1 dollar. So it was possible to buy 7 million tickets for 7 million dollars to ensure a jackpot of 27 million. A consortium of 2500 small investors aimed to raise the money to buy all these tickets. However, they managed to buy only about 5 million tickets. They still managed to win, (See: The improbability Principle — David Hand, page 120)

To buy 302 million tickets is a different story. First you need to have 604 million dollars in cash. Second, you need to actually to buy all these tickets, and you have only 4 days to do all these shopping. In 4 days there are 345600 seconds, so you need to buy nearly 900 tickets per second, while you make sure that each ticket carries a different combination of numbers. The logistics may be difficult. In 1992, that consortium mange to but only 5 million tickets in about a week. You may not be able to buy all the tickets you need.

Second, when you win, you will probably want the money in cash, and not in annuity payments. So, the 1.6 billion reduces to 910 million, and the government will take 25% tax (and more later). You will end up with 684 million. Still a net profit of 80 million dollars. Not bad.

Third, it is possible that you will share the jackpot with someone else, maybe even more than one person. As we already saw, there is a good chance for this scenario — if 210 million tickets are sold then the probability for sharing the jackpot is about 50%. Even if you share the jackpot with just another person, your share is just 800 million, and if you want to cash it it will shrink to 342 million, a net loss of 262 million. That’s not good.

And finally: should you buy a ticket? Yes, buy one ticket. It’s fun. And for two dollars you buy the right to hope winning the jackpot, as Prof Robert Aumann, the Nobel Prize winner said.