I am reading, and quite enjoying, *How Not to Be Wrong: the Power of Mathematical Thinking *by Jordan Ellenberg, a professor of mathematics at the University of Wisconsin-Madison.

The book as a whole is excellent. He misses the mark, at least a bit, when he tries to take on Adam Smith on the topic of lotteries.

Smith wrote

There is not, however, a more certain proposition in mathematics, than that the more tickets you adventure upon, the more likely you are to be a loser.

Ellenberg points out, quite rightly, that if you buy two tickets, you are only half as likely to be a loser than if you buy one. He uses as an example a lottery of ten million $1 tickets with a $6,000,000 payoff, and makes a big deal about an inflection point at 6 million tickets purchased, where you go from a possibility of winning to a certainty of losing. He says it's hard to reconstruct Smith's reasoning.

On the contrary, while Smith's sentence is incorrect, his reasoning is blindingly easy to reconstruct. As anyone who has read him knows, Adam Smith was inseparably grounded in the practical and concrete. What is the practical effect of buying more lottery tickets? In fact, given that there is no such thing as a fair lottery, buying more lottery tickets in practice means losing more money. This is a simple application of the concept of expected value. Ellenberg actually introduces expected value on the page after he takes Adam Smith to task, but he misses the opportunity to illustrate the key fact about lotteries.

Adam Smith would have been completely right with a minor change in word order. Instead of writing

the more tickets you adventure upon, the more likely you are to be a loser.

he could have written

the more tickets you adventure upon, the more you are likely to lose.

In Ellenberg's example of a lottery with ten million one dollar tickets and a six million dollar payoff, the expected value of one ticket is the six million dollar payoff divided by the ten million chances to get it: in other words, 60 cents. So each ticket has an expected loss of 40 cents. The more you buy, the greater your chance of winning, but also the greater your probable loss.

Here is the proof in four lines or R code, based on Ellenberg's example:

# ten million tickets, each worth one dollar

tickets = 1:10000000

# our expected winnings, at each number of tickets purchased

wins = 1/10000000*tickets*6000000

# our expected profit or loss

net = wins-tickets

# what does it look like

plot(tickets,net)

The graph of expected profits is a straight line down from roughly zero to minus 4 million dollars. Ellenberg actually speculates that Adam Smith may have mistaken a curve (the probability of making a profit) for a straight line, but Ellenberg seems to have missed the straight line that was string him in the face.

The more you invest, the more you are likely to lose. That, as Adam Smith surely understood, is what should drive the analysis of lotteries as investments.