A roulette player is down to his last few chips. He’s missed his lucky number at least 36 times in a row. Surely Lady Luck owes him one by now.
All those chips go on 17. Yet the ball lands on 35, for the third time in the last ten spins. Now broke, the player heads to his car to drive home, confused and frustrated. Does the Universe hate him? Was the game rigged?
Anyone who’s done much gambling has experienced losing streaks. They’re not much fun. In fact, they feel downright unfair.
Even if you’re not prone to suspicions of rigged games or bad karma, you might find yourself wondering: “What are the odds?”
There are all sorts of losing streaks, from crapping out immediately on your last three turns with the dice, to losing your bankroll endlessly doubling down at blackjack, to being a lifelong lottery player who never wins a big prize.
All of those have their own probabilities, and you’d have to calculate each one separately. However, there’s one very specific and common type of losing streak for which we can provide a general rule.
Let’s call it The Mystery of the Overdue Win.
When Is a Win Overdue?
Let’s go back to our fictional – and unlucky – roulette player. Assuming he’s playing American roulette, with a 0 and 00, there are 38 numbers on the wheel. You don’t need a PhD in Statistics to know that this means the odds of the ball landing on his number are one in 38.
Our gambler knows this too, sort of. But as he approaches 38 spins without a win, he feels that his number is “due.”
That’s called the Gambler’s Fallacy. And it’s called that because it’s false. If it were true, it would be impossible to flip Heads twice in a row on a coin; after the first Heads, Tails would be “due” and streaks would never happen. And yet, we all know it’s entirely possible to flip Heads (or Tails) repeatedly.
Over the very long term, one in 38 spins will produce whatever number you’re hoping for. However, the odds on any given spin never change. One in 38 is about 2.63%. Those will be our gambler’s odds of getting one on the next spin, regardless of whether he just hit a 17, or hasn’t seen one in 100 spins.
This is basic casino math, and important to understand if you want to gamble responsibly.
It still doesn’t answer our question, though, which was how likely was it for our gambler to be this unlucky?
The answer is that this should happen to him a little more than one-third of the time.
The Basic Math of Probability
If your math classes included an introduction to probability, you may already know where we’re going. If not, buckle up, because it’s about to get nerdy in here. But don’t worry, I’ll try to keep it simple.
There are really only two principles we need to know to get the answer we’re looking for:
- When you want to know the chances of two things both happening, you multiply the odds together. (Example: the chance of a coin flipping Heads is 50%. Therefore, the odds of flipping Heads twice in a row is 50% x 50%, or 25%.)
- The odds of something happening are the same as 100% minus the odds of it not happening. (Example: if you don’t flip Heads twice, that implies that you flipped Tails at least once. So the odds of getting Tails at least once in two tries are 100% minus 25%, or 75%.)
So, let’s say we have a one in “X” (that is, 1/X) chance of getting the result we want, and we try X times.
Based on Rule 2 above, the odds of missing once are 1-(1/X), which you can also write as (X-1)/X. For instance, the odds of getting our number in American roulette on a single spin were one in 38, so the odds of not getting it are 37 in 38.
If we try 38 times, Rule 1 says that the odds of missing every time are 37/38 multiplied by itself 38 times. That is, 37/38 to the 38th power, which you can write as (37/38)38.
That happens to be about 36.3%.
Is it Always the Same Chance?
If you try the same trick with other scenarios, you’ll find the numbers differ a bit. This is especially true with relatively high probabilities and small samples.
For instance: we already said the chance of failing to flip Tails twice in a row (that is, flipping Heads twice) is 25%. Missing a one-in-three chance three times is (2/3)3 or 29.6%.
Here are the next few:
- Missing four one-in-four chances: (3/4)4 = 31.6%
- Missing five one-in-five chances: (4/5)5 = 32.8%
- Missing six one-in-six chances: (5/6)6 = 33.5%
Already, you can see that those numbers are creeping closer to what we have for roulette. Does it keep going up forever or does it stop somewhere?
Let’s look at the odds of missing a one-in-a-thousand shot, one thousand times in a row: (999/1000)1000 = 36.8%.
That’s only slightly more than roulette. And, in fact, even if you were to calculate one trillion one-in-a-trillion chances, you’d get a very similar answer.
The Really Nerdy Part
If you just wanted to know the number, you can stop here. But if you have some curiosity about math, you might be interested to know that what we’re closing in on is a number called e, or rather 1/e.
What is e? It’s a special number in math, like π, which we all learned about in high school. Except that instead of having to do with circles, it has to do with exponential functions. The formula we’re looking at here is exponential, so it’s no surprise to find e cropping up.
When you have a formula that produces increasingly similar results as you increase the variable (“X” in our case), that’s called a convergent sequence. Here, the sequence converges on 1/e, which you can verify using Wolfram Alpha (which is like Google for numbers).
Like π, e is irrational, meaning it goes on forever and never repeats. It starts off like this: 2.718281828459…
And 1/e is 0.3678794411714… which, if we express it as a percentage, is about 36.8%. So, for relatively small chances, like spinning a number in roulette or winning a slots jackpot, you’re going to be about 63% likely to hit it within the “expected” timeframe, and 37% likely to miss due to an unlucky streak. (Those chances of missing are offset by the possibility that you’ll hit it multiple times.)
Are Wins Ever Actually Due?
There’s one important condition to keep in mind here. All this math only works if each attempt really is independent. That’s true for most things you’ll see in a casino:
- Dice rolls
- Roulette spins
- Slot machines
- Reshuffled decks of cards
However, it’s not always true for everything. For instance, an instant lottery guarantees a certain number of each prize. If half the tickets have been sold and no one has won the top prize, winning it is now twice as likely as it was to begin with.
In a brick-and-mortar casino, the most common place you’ll see non-independent probabilities is in blackjack. Until the deck is reshuffled, the chances go down of seeing cards you’ve already seen frequently, and they go up for cards you haven’t seen as often. This is what card counting is all about. (Don’t try it with online blackjack, though – the virtual cards are “shuffled” after every hand.)
That doesn’t mean that losing a lot at blackjack makes you due for a win. It does, however, mean that if you haven’t seen many face cards, say, they’re more likely to come up in future hands. Card counting is extremely difficult – and likely to get you booted from the casino – but the basic principle is that a deck with lots of high cards left in it favors the player, while one with lots of low cards favors the house.