A version of the Samuelson Bet, named after the economist Paul Samuelson is as follows:

A single coin flip, heads you win $1000, tail you lose $500. Do you take the bet?

Most can see that in this bet, you are expected to win more than you lose. The expected value, matter of fact, is +$250. In other words, you are expected to become $250 richer each time you play this flip. However, with a potential $500 loss looming over your shoulder, other than those with deep pockets or a taste for gambling, most would shy away from making this bet.

Now imagine

Same bet as above, but we flip 1000 times. Each time you either win $1000 or lose $500. Would you now take the bet?

I cannot imagine anybody that would turn down that bet. But what changed?

What changed was not that you’re any less likely to lose each hand. The odds remains at 50/50 per hand.

What changed was as the number of attempts was increased significantly, the odds of you making money increases dramatically. Intuitively, the bet is a good deal, because the odds of winning and losing are even, and you’d win twice as much than you’d lose. Yet in one single play, some of us wouldn’t take that bet, because the odds of losing a significant amount of money ($500) is still 50%. When we expand the number of plays, however, it seems like the bet is looking more and more attractive as the effects of variance is greatly reduced. In a single flip, the result can go either way, but in 1000 flips under a normal Gaussian distribution, the standard deviation (sigma) becomes 15.81. So in 1000 flips, one can predict with an accuracy of ~69% that the coin will land somewhere around 500 heads and 500 tails ±16 (484H/516T ~516H/484T.) We can increase our prediction to ~95% by going 2 standard deviations away, meaning there is about a 95% chance that 1000 coin flips would land between 468H/532T ~532H/468T. This lead to the interest discovery that in order to NOT win money in the 1000 coin toss version, we would have to flip twice as many tails as heads (333H/667T). That is an event more than 10 standard deviations away, also called a 10 sigma event (1 in 1.5265*10^23). This means the odds of losing money is basically 0. By comparison, in a 10 flip trial, the odds of losing money becomes only 2 sigmas (1.58) away, which gives us around a 5% of losing money.

Danny Kahneman, a psychologist and Nobel economist, has an interesting take on the Samuelson bet in his book Thinking, Fast and Slow. He suggests that many of the risks in life should  be framed as a series of much less obvious Samuelson Bets, each with a much lower but still positive expected value (Something like heads you win $100, tails you lose $95.) Rather than narrowing viewing each opportunity as a singularity and obsessing over the potential losses in each case, we can take a broader frame of the situation. Just like in the Samuelson bet, although each individual bet will seems risky, the aggregate outcome of taking many smart and calculated risks in life will result in a net positive. It’s a wonderful message that I think all of us can take some time to ponder over.