Welcome to the first issue of Gambler’s Gain—a weekly newsletter that brings you a selection of news and math related to gambling.

Gambler’s Gain is put together by the team at BtrBet, who are also building tools you can use to bet better. And if it’s more your style, you can find us on Twitter at @btrbetapp.

News

Special Olympics World Games Odds

BetOnline.ag is now offering odds in twelve competitions across the Special Olympics World Games for track and field, basketball, cycling, powerlifting, soccer, and swimming. The company’s aim is to generate publicity for the competitions, and it’s in line with BetOnline’s larger goal: offering odds on any sanctioned competition with two or more participants.

This seems like a winning strategy for everyone involved—more awareness for the Special Olympics, and BetOnline gets publicity and their vig from bets placed on the games.

I can personally speak to the influence betting can have on awareness: a few weeks ago, I couldn’t name a single WNBA team. After betting on league games for the past month, I know every single one. The simple lever to raise my awareness? Sportsbooks offered odds on contests for the sport.

Gambling with hookworms

Dog racing is rife with physical and ethical ills, and there’s data to suggest unexpected side effects for everyday pet owners.

A series of studies published over the past few years by parasitologist Ray Kaplan and colleagues suggest that the greyhound racing industry plays a key role in the evolution of drug-resistant hookworms. As betting on dog racing has declined, more organizations are working to get racing dogs into loving homes—a good thing! But this also means that drug-resistant hookworms are working their way into domestic pet populations.

This isn’t to say that more betting on dog racing is any kind of solution to this problem, but it does suggest that as gambling becomes more available and present in people’s lives—and resources shift to and from various industries—we’d be wise to think through its knock-on effects, which might not always be obvious.

Math

Betting on 6

Gambling predates recorded human history. The mathematical study of probability, however, can be traced to the 1650s, when the Chevalier de Méré asked his friend Blaise Pascal to provide an explanation for an observation he’d made: he could typically win by betting that he’d roll at least one six, in four rolls of a six-sided die; however, the same could not be said for betting that a pair of sixes would be rolled, in twenty-four rolls of two dice. Today, let’s only think about rolling a single die many times instead of rolling a pair of dice.

As Edward Packel writes in The Mathematics of Games and Gambling, “In a series of letters between Pascal and Pierre de Fermat, de Méré’s difficulty was explained, and in the process, the idea of probability, the eponymous and Pascal triangle, and the ubiquitous binomial distribution emerged.”

Let’s get into the math

It’s fairly simple to break down the reason that betting on rolling at least one six in four rolls is typically a winning bet.

The chance of rolling any number other than a six on a given roll is 5/6. This means that the chance of rolling a number other than six on four successive rolls is 54/64, or 625/1,296. This is because each roll is independent of the other rolls, so the probabilities to roll a non-six on each roll are multiplied:

(5/6) x (5/6) x (5/6) x (5/6) = (5 x 5 x 5 x 5)/(6 x 6 x 6 x6) = (54)/(64)

Let’s take it a step further: This implies that the chance of rolling a six in any one of those four rolls is 1 - (625/1296), or 671/1,296. And because 671 is more than half of 1,296 (51.77%), a bet on six is more likely to win than to lose.

Doing a similar set of calculations for a varying number of rolls yields an interesting result.

# of rolls	chance of rolling a non-six on every roll	chance of rolling a six on at least one roll	probability of rolling at least one six	true odds	expected profit (loss) on a $1 wager
1	5/6	1/6	16.67%	5:1	($0.67)
2	25/36	11/36	30.56%	25:11	($0.39)
3	125/216	91/216	42.13%	125:91	($0.16)
4	625/1296	671/1926	51.77%	625:671	$0.04
5	3125/7776	4651/7776	59.81%	3125:4651	$0.20
6	15625/46656	31031/46656	66.51%	15625/31031	$0.33
7	78125/279936	201811/279936	72.09%	78125:201811	$0.44
8	390625/1679616	1288991/1679616	76.74%	390625:1288991	$0.53

Seeing the numbers this way makes de Méré’s observation seem obvious, and makes it clear that four rolls is the fewest number of rolls for which this bet could be expected to win more than it loses. It’s also the smallest number of rolls in which there’s an expected profit based on the true odds.

It would be interesting to see a casino game made where the player’s payout is based on how many rolls it takes them to roll a six. In such a case, the casino could offer house odds that look simpler, but that ensures that the bettor is always expected to lose.

If you were making this casino game, what house odds would you offer? Reply with the house odds you’d give and next week, we’ll share what we would offer…and talk about the futility of betting on roulette, even when betting under the true odds. (A little teaser: the true odds for betting on red on an American roulette wheel are 20:18, but typical house odds are given at 1:1).

Product updates

Are you a sports bettor? If so, we want to hear from you! We’re curious how sports bettors track their bets and what friction you run into when placing, tracking, and hopefully celebrating your sports bets. Let us know what works or doesn’t work for you at zach@btrbet.app or by replying to this email.