Two summers ago at church, I played a game with Jack Ding.
Bored at the church youth group’s ice-breaker introductions and the “beginning of the year” stuff, I pulled out a penny and showed it to Jack.
“Heads or tails, bet a quarter?” I asked.
“Sure. Heads,” Jack said.
I tossed it to the air. The coin flicked, spinned, glistened, and wobbled before finally landing on my legs. When it did, it was tails.
“Wait — wait, go again!” Jack demanded.
So it was like that, at the corner of our church’s gym, that we started this game of chance. Initially, I was pretty lucky, and I won a couple of dollars. As I won more and more, Jack demanded higher stakes; a quarter turned into fifty cents, and fifty cents turned into a dollar.
And it was then that I started losing.
For around 4 coin flips in a row, the coin landed heads, and I called tails. At one dollar per flip, my net gain quickly dropped to zero, before it started going negative.
“No way,” I said to Jack, frustrated, “go again. Tails for two bucks.”
Jack agreed. I tossed the coin. It was heads. At this time, I owed him three bucks.
“Nah, nah, nah… go again! Three bucks!”
Jack agreed. I tossed the coin. Heads again.
“Bruhhh! go again! Five bucks!”
Jack agreed. It was heads again.
“Ten bucks! go again!” I said.
“Nah, let’s do five,” Jack replied.
“Eight!” I bargained.
“Five,” Jack insisted.
“7.5 or nothing,” I bargained again,
“Nothing then, a tail is coming anyways” Jack said.
“Sure let’s do five, ” I subdued, losing my bargain due to desperation of winning money back. But luckily, Jack agreed and didn’t actually stop playing. Yes. A tail is coming. I called tails.
But it was heads again.
At this point, I owed him 16 bucks already. But am I going to quit playing? Heck no. I knew that tail was due. It has been 8 heads in a row, and I knew that it is statistically very unlikely (an 1/256 chance!) for this to happen already. One more heads, and it would be a 1/512 occurrence. And am I going to root for that 1/512 occurrence to not happen to me? Of course yes.
“Six bucks, go again,” I said to Jack, and he agreed.
And… heads again.
“F—,” I thought to myself.
“Last round! 20 bucks!” I told Jack, thinking that I will definitely win this one. It has been 9 heads in a row, and the chance for 10 heads in a row less than 1 out of 1000. Am I going to be THAT unlucky? Of course not. A tail is definitely coming.
To my surprise, Jack took the offer. I tossed the coin — and can you guess heads or tails?
I’ll wait.
Ready?
It was heads.
Confused, frustrated, furious, and in debt, I threw that “cursed penny” into a trash can and raged quit. How am I this unlucky? Ten heads in a row? What the hell?
Well, looking back at this experience two years later, I could affirm that I was unlucky; but I also fell for a dangerous illusion and made a serious mistake that I hope you will never make after reading this post — the Gambler’s Fallacy.
See, we humans have a tendency to believe that things will even out. If the chance of flipping tails is 50 percent, we expect that tails will be flipped 50 percent of the times — and when it doesn’t, we think that tails are going to land more frequently so that the numbers will even out, or that they are “due”.
This is why gamblers like to “double or nothing” after they lose a bet. They believe that the chance of them consistently losing are so low that they will make consistent profit if they just keep doubling their bet after they lose, as they will eventually make their money back plus the profit they intended to win originally. For example, when a gambler who wanted to win $1 from betting $1 loses his first bet, they would bet $2; and if they lose again, they bet $4; lose again? No problem! Bet $8; bet $16; bet $32; bet $64; bet $128… They believe that the Universe owe them a win, and if they keep betting, they will eventually win back everything plus the $1 they originally wanted.
Pretty dangerous mindset, huh? This strategy of “doubling after losing” is so notorious that it has its own name — the Martingale System, and it was responsible for robbing away countless people over hundreds of years of every last dime they had.
See, when people who fall for the Gambler’s Fallacy thinks they are “due” for a win, they don’t care how much they lose before the Big Win hits. However, they forget that probability has no memory, and the outcome of each future bet has no relationship with the outcome of the bets in the past.
This means, after they lose, gamblers are essentially tricked into wagering exponentially more into a game that gives them the same odds. And in chasing after their win that is “due”, they risk sums of money that they cannot afford to lose.
And everything above assumed that the game is a coin toss, a game of even odds, when in reality, the odds are almost always against the gambler.
When you are playing a game of negative expected value (which is almost always the case with gambling), you would should expect to lose some money every time your play it, and the amount you lose is directly proportional to the amount you bet. However, people who fall for the Gambler’s Fallacy usually ends up betting enormous amounts, meaning that for each large bet they make, their expected value is enormously negative. They go broke way faster than your average quitter.
Wonder how they built these beautiful hotels at Las Vegas?
So hold your chips guys, do not chase after your losses.
Or better yet, do not gamble.
Thanks for the advice! I found your blog to be really interesting. My favorite part was when you did such a great job of creating such suspense and anticipation about the result of the final coin flip. I feel like the way you wrote that really wrapped readers, like me, to continue reading in order to see the results of the bet. One minor note I have is that at the beginning of your blog, with all of the dialogue between you and Jack, I found it a little difficult sometimes to follow who was talking, and when. But other than that, I feel like you did a really great job of engaging the reader, using a very entertaining anecdote, and sufficiently explaining a concept to give advice! I thought that the topic you picked was really interesting, I had an idea of what the Gambler’s Fallacy was but I had no idea it was a real, somewhat proven theory. I think your explanation of it was helpful in explaining the idea to both people that knew a little bit about it as well as people who have never heard of anything like this before.
I like the narrative and personal experience with Jack Ding that started the post was a nice touch. Then you express the lesson learned. This was an interesting read. Thanks for sharing Frank.
I found your blog to be very intriguing, I was always told to stay away from risking money as a kid and it seems like your reasoning proves why, and it’s simply just math. Also the reasoning that the house always wins is uncontested, they rack in ridiculous amounts of money just because gamblers think they can make up their losses by risking more. Additionally, Casinos will offer expensive drinks, put no clocks in the room so people lose track of time, and have cruddy carpets to get people’s attention on the tables. One thing that I think is better is betting, because at least with betting there are measures and ways for individuals to get an edge, and that it’s not pure luck. However, there are two reasons to bet, for money and for fun. And I’ve gotta say, putting 50 cents on a 10 leg parlay for $200 sounds like fun.
Holding chips? What? I could never.
Loved the narrative bro, but I do want to go into the mathematical aspects of the gambler’s fallacy. Suppose we’re playing a game that pays our bet with a probability p, where 0<p<1. If we go double or nothing each time, then that means that for a bet valued at $x can lead to either a gain of $+x or a loss of $-x. The expected gain is then x*p-x*(1-p). If p<0.5, then this is negative.
But what if you did continuously bet until you win? In your example the coin flip, doubling the bet each time could actually lead to a NET PROFIT if you were persistent enough. After all, if you start off with a bet B, and double it each bet until you win, then your net gain after n such bets will be 2^n-B(1+2+4+…2^(n-1)) = B*2^n-B(2^n-1) = B. And the probability you win at some point?
100%
Maybe the moral is if your friend is persistent, then keep at it until that impending win – assuming infinite time and trials. Or? Just guess heads next time!
Holding my chips tightly,
Kai Liu
Kai, I don’t know if your comment was satirical or not, but it perfectly demonstrated the gamblers fallacy mindset — that you can consistently profit if you just keep doubling.
When you do math about house games, you should only and always think in terms of two things: expected value (EV) and variance. EV determines whether the game is profitable, variance determines whether the game is safe.
For the example of a coin toss, you have even odds, which means the expected value of any bet would always be 0, regardless of bet size B. It’s not possible to have a “net gain” in games like that. “Doubling after losing” only increases your variance as it lures you into wagering more money per bet. This means you are risking more money on a 0 EV game every time you double. Since you only have a limited bankroll, chances are sometimes you lose 10 times in a row and you lose everything. Conversely, if you win 10 times in a row, you only win 10B.
Mathematically, a coin toss should give you 0 net gain, but in reality, when people double after they lose, they can lose everything on an unlucky streak due to the high variance of their strategy.
Betting is something that I also have found quite interesting, even when I was younger and had no actual money to bet. Instead of with coin tosses, we’d do over unders. Over under the speech lasts this many minutes or over under some number on a dice roll. Whatever it was, there was pride and fun in winning, even when money wasn’t on the line. As you said in your blog, there is this belief that we are “supposed to win” after a certain amount of losses, and thus we can take unnecessarily high risk for little reward. Casinos having drinks, lacking windows, and having other incentives that make you stay while losing track of time makes betting even more dangerous. That chance that you “win big” is certainly enthralling, but most of us will end up like you, unhappy and $40 down.
Frank, this is a really intriguing blog post! I haven’t partaken in betting of any kind, and your insights make me really want to stay away because it seems easy to fall victim to the Gambler’s Fallacy. Your topic seems well-researched, and I like how you introduced it with a suspenseful dialogue and then proceeded to lay out the facts. This theory reminds me of the self-serving bias that we learned about in AP Psychology. It states that we attribute positive events to our actions, and we attribute negative events on external factors that are out of our control, and this can be seen in sports, games, or anything that involves some sort of competition. The self-serving bias might influence the behavior of gamblers too, encouraging risky actions as they win more and have a higher self-esteem. Overall, this blog post taught me something I knew nothing about, and reinforced my fear of betting (which is a good thing).