What are the chances that the lottery ticket that you’re holding in your hand is a winner? This sort of question is an applied probability problem. With millions of dollars on the line, it would be good to have a solid answer. What are the odds of winning the lottery?

## How Does The Lottery Work?

To answer this sort of question we need to know how a lottery works. In lotteries, a person plays by buying a ticket with a choice of numbers from a certain range. At a specified time the organization running the lottery randomly generates numbers from this same range. The grand prize, sometimes worth several millions of dollars, is awarded for matching all of the numbers. In some lotteries, lesser amounts are paid for matching all but one or two of the numbers.

Those are the generalities, but we need to know the specifics of the game that you purchased a ticket for. These specifics will calculate the exact probability that you have of winning.

## A Sample Game

One game, known by various names such as Daily 4 or Pick 4, involves choosing four numbers from 0 to 9. The order of these digits is important, so 1234 is a different choice of digits than 1243 or 1324. The probability of winning this lottery is given by determining the total number of four-digit numbers possible. Since each of the numbers is chosen independently and there are ten choices for each, the total number of four digit numbers is 10^{4}=10000.

This means that the probability of winning is 1/10000 = 0.01%. Games of this sort typically do not pay that much and are not what people associate with winning the lottery. A typical payout for a $1 bet on a lottery such as this is $5,000. While this sounds good -- who wouldn‘t want to multiply their money by 5000 -- realize that on average you would have to play thousands of times to make winning likely.

## Another Lottery Game

Another type of lottery game pays more if you win it, but it is much harder to win. An example is where six numbers are chosen from 1 to 48. Here the order of the numbers is not important, and so we need to calculate a combination. We can choose six number from 48 in a total of *C*(48, 6). By the combination formula this number is 48!/(6!42!) = 12,271,512. As matching all of these numbers perfectly accounts for one of these combinations, the probability of matching and winning millions is 1/12,271,512.

## How Likely Is It?

So those are the numbers, is there any good way to interpret them? Let’s look at the millionaire game and its probability of winning at 1/12,271,512. Winning this is very unlikely. To be assured of a 50% chance of winning you would need to buy over eight million different tickets. The number 12,271,512 is roughly that of the population of the entire metro area of Los Angeles, California. So the probability of winning the lottery is the same as running into a particular person, chosen in advance while walking the streets of LA.

Another way of looking at this is to look at other probabilities. Numbers are somewhat hard to come by, but it’s been estimated that 100 people a year in the U.S. are struck by lightning. With the current population at 307 million, the probability that you will be struck by lightning this year is 100/307,000,000 = 1/3,070,000. So you’re four times more likely to be struck by lightning than to win the lottery.

Sure, some people win millions in the lottery. It’s just very likely that it’s not going to be you.