If I want to increase my profit for an existing product I have one of two ways of doing so: raise the nominal price, or reduce the size of my product. If a candy company currently prices its candy bar at $1 for a 6 oz. bar, it can increase its profit by raising its price to $1.20 for the 6 oz. bar, or it can instead keep the price fixed at $1 and decrease the size of the candy bar from 6 oz. to 5 oz. It effectively did the same thing, but the latter method is a bit more surreptitious than the former. (Selling 60 oz. of chocolate would increase total revenue from $10 to $12 in both cases.)

On January 15, 2012, the Powerball multi-state lottery increased the purchase price of one of its tickets from $1 to $2, leaving unchanged the odds of winning and the size of the jackpot.

On October 19, 2013, the Mega Millions multi-state lottery game also raised its price, but did so more like the example of decreasing the size of the candy bar—they decreased the odds of winning while maintaining the $1 price per chance.

Before October 19th, there were 175,711,536 different number combinations for Mega Millions, of which only one wins the jackpot. With the new Mega Millions, there are now 258,850,890 different combinations. To put this change in perspective, prior to October 19th, winning the old way was equivalent to lining pennies edge-to-edge on a driving route from Springfield, MO all the way to Seattle, WA. One of those pennies was marked with an X on the underside and you had one chance to drive to some location on the route, stop at one of these pennies and turn it over. If that penny had the X you won the jackpot. Winning with the new Mega Millions game extends the route all the way from Seattle to Jacksonville, FL. Good luck finding that one penny!

Decreasing the probability of winning was partially offset by increasing the starting jackpot for the winner and by how much the jackpot increases each week there is no winner. So what really matters here is the expected value of buying a chance at winning the Mega Millions jackpot. Because the jackpot does change, and because more people play as the jackpot increases, increasing the probability that the winner will have to share the winnings, I will use just the lowest jackpot amount, which is what the jackpot begins with until someone matches the jackpot numbers.

There are nine possible ways to win money from purchasing any one number combination, but winning is not cumulative. For example, if you correctly picked, say, 3 white balls and the power ball, you don't win the jackpot, but you do win $50. However, you don't also win any lesser prizes for correctly having combinations requiring fewer numbers. You don't win an additional $5 for having three white balls, and $5 more for having 2 balls, etc. You simply win $50. Therefore, the value of any given number combination is equal to the maximum expected value of winning any prize with that combination, which for both the new and old games is winning the jackpot.

As seen above, the expected value of winning the jackpot for the old Mega Millions game for the lowest possible jackpot is 6.83¢, and for winning the new game it's 5.79¢. Did you get that? The expected value of playing (I hate using that word to describe buying a ticket) the Mega Millions game, which still costs $1, decreased by 1.03¢ as of October 19, 2012. That's equivalent to a roughly 16% increase in the price of buying any one number combination. And most people who play probably didn't even know it. (Then again, if you buy lottery tickets you don't understand probability theory either.)

This is not the first time that Mega Millions surreptitiously increased the price of playing its game. As shown in the chart below, and graphed below that, the expected value of a Mega Millions number today declined nearly 60% of what it was at the inception of the game (then known as The Big Game) on September 6, 1996.

But don't worry, it's all going toward improving education, at least in North Carolina.