## 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?