I recently had a disagreement with a friend about what it means for an event to have a probability of happening every X times.
Take for example a coin that if flipped has a 50% probability of landing heads and a 50% probability of landing tails.
My understanding is that while the probability of landing tails 3 consecutive times is smaller than that of landing it 2 consecutive times, if in the last two flips you got tails it has no effect on the probability of you getting tails in the following coin flip.it stays 50-50.
But he seems to believe that the probability of you getting tails gets smaller and smaller the more you land multiple consecutive tails.which sounds very counter intuitive to me.
For a more concrete example, he seems to think that if an event has a probability of occuring every 10 years, and it’s been more than 10 years since the last time it occured, we are living on borrowed time and the probability of it happening now is extremely high while I think it had the same probability of occuring today than it had of occuring 9 years ago.
Which one of us is wrong, please give examples.
In: 5
You’re exactly right. Your friend has fallen victim to something called the [gambler’s fallacy](https://en.wikipedia.org/wiki/Gambler%27s_fallacy).
So while it is very unlikely, for instance, for a coin to land heads 100 times in a row, it does not make it more or less likely to land heads on the 101st flip.
>For a more concrete example, he seems to think that if an event has a probability of occuring every 10 years, and it’s been more than 10 years since the last time it occured, we are living on borrowed time and the probability of it happening now is extremely high while I think it had the same probability of occuring today than it had of occuring 9 years ago.
This is harder to answer because “probability of occurring every 10 years” isn’t well-defined. It could mean a lot of things. But I think you’re talking about a situation like this: lets say that there’s a volcano, and scientists estimate that every year it has a 10% chance of erupting. A naive interpretation might say that it will go off every ten years. But in fact, if we do the calculation we’ll find that there’s roughly 35% chance that ten years will pass with zero eruptions. If that happens, you’re correct: the probability has not changed, and there is no “borrowed time.”
Your concrete example is actually less concrete than the coin. It depends on what the process that generates the event it. e.g. If a volcano usually erupts every 10 years, and it’s been 12, then that eruption is much more likely to happen. In that cases, it’s because the event (eruption) is the culmination of a process (geological stuff that I know basically nothing about), and if the event hasn’t happened, the underlying cause it still accumulating.
Your coin flip on the other hand has no such process. You simply decide to flip and then do. That makes the coin flip you’re doing right now completely independent from the prior flips, so your next flip is 50/50 no matter what the prior ones looked like.
On a fun side note here, if your friend really wants to cling to this, offer to play a game with them. You’ll start flipping a coin; at any time they can stop you and say that they are willing to bet on the next flip. If they’re right you pay them $3; if they’re wrong they pay you $4. If your friend’s model was correct then they could easily make money from you by just waiting for 2 in a row and then betting the opposite. However, your friend’s model is incorrect and you’ll take them to the cleaners. I worked a similar plan against a buddy in college while demonstrating the Monty Hall Paradox; I bought a cheap 6 pack and gave him the rest back after. 😛
The coin flip and the “every 10 years” might be different things and the same rule might not apply to both of them.
For the coin flip you’re right because all that matters is the *unknown* coin flips that haven’t happened yet. Once some of the flips are in the past, they no longer affect the probability of the next flip. If you *don’t know yet* what the 3 flips are, then 3 flips are only 1/8 likely to all be heads. But if you *already know* that 2 of them are heads, then the remaining one flip is a normal flip like any other and it’s 50/50. Essentially, the fact the 2 of the flips are now in the past and have already been heads, that has set their probability of being heads to a constant of 1. Instead of (1/2) * (1/2) * (1/2) you now have 1 * 1 * (1/2).
But the event with a probability of “once every 10 years” might not be like the coin flip. It could be something that gets more likely the more “aged” it becomes, and 10 years is just the mean time between events. For example, failure of parts due to fatigue is like that. “Mean time between failure” is often caused by the wear and tear of age, so something being 8 years old is different from it being 1 year old. Another example is a geyser that goes off, say, once every 2 hours on average, but with some variation. It goes off when pressure builds up to the breaking point, so the longer it’s been building up the closer to the breaking point it has become.
The coin toss is really a special case scenario because the age of the coin is irrelevant, the time spent so far is irrelevant, and so there’s no means for the past events to alter the future events.
It’s important to note that this is assuming you *already know* the coin is a ‘fair’ coin, rather than a ‘loaded’ one. If you don’t yet know that and you are *testing* to find that out, then things are different. If there’s a chance the coin could be unbalanced and biased, then a past pattern of many many heads in a row starts to increase the odds that that is the case. (Although, that wouldn’t mean it’s ‘due’ for tails, just the opposite. It would mean you’ve established that it’s likely to have a bias making it less likely to be tails.)
You are right about the coin. If you flip two heads, the third is still exactly 50-50. This is because every coin flip is independent of the others.
Your concrete example is more complicated and depends on your assumptions, and for practical purposes your friend is probably correct. If a geyser erupts every 10 minutes and 10 minutes has passed, it is much more likely the geyser will erupt in the 11th minute than it was in the 1st. This is because every minute that passes increases the likelihood of eruption. The eruption rate is *dependant* on time.
There really is no answer to your question without answering the question of if events are dependent or independent of one another
It depends where you start the measurement.
Each flip is independent of all the others and has a 50/50 chance assuming the coin is “fair” and that the position facing upwards when you flip is random.
The odds of 1 tails is 1/2, 2 in a row is 1/4, and 3 in a row is 1/8 and so on. But if you asked the odds of any singular flip being tails, it’s 1/2 regardless of what happened before.
A few questions for your friend that might help, roughly in order you should ask them:
If you flip a coin, and it lands on tails, according to them, the odds of getting heads next time is larger. But what if you flipped it in secret, and all those flips landed on tails?
What if you flip coins, but don’t keep track? Does that have an influence?
Does the coin being flipped have a memory of all the outcomes of its previous flips? It would need one in order to keep track of how many tails it got in the span of its entire life.
If you drop a coin, does that count as a flip, or does there have to be intent?
What if you flip it, but catch it before it lands?
>My understanding is that while the probability of landing tails 3 consecutive times is smaller than that of landing it 2 consecutive times, if in the last two flips you got tails it has no effect on the probability of you getting tails in the following coin flip.it stays 50-50.
Your understanding is correct here. The thing we have to keep separate is the *probability of one thing happening,* and the *probability of multiple things happening.* Let’s pretend I’ve just flipped 3 heads in a row. What are the odds of my text flip being heads? 50%, because that’s just the odds of *one thing happening*. But what are the odds that I flip a coin 4 times and get all 4 heads? Well that’s only 1 in 16, or 6.25%, because that’s the odds of *each* of those coin flips put together. So you have to distinguish if you’re looking at just one event, or the sum of all of the events. The odds of a roulette wheel landing on 15 are 1 in 38, and the odds of landing on 15 twice in a row are around one in 1400. But if it’s already happened once, you can ignore that part of the odds, they’ve already happened – the odds of the next spin is still just 1 in 38.
Another way to think about it is to imagine a poker hand – imagine that you have a royal flush 10-J-Q-K-A all of one suit (let’s say clubs, for the example), that’s the rarest possible hand, right? Well now imagine that you have a 2 of spades, a 6 of diamonds, an A of hearts, a Q of diamonds, and a 10 of clubs. Well that’s a really common hand, right? Actually, it’s *exactly as rare* as the royal flush – both hands have the same seriously tiny chance of being dealt to you. We just see the good hand as significant and the trash hand as insignificant, so we don’t think of it that way. But it’s the same with many random events – the odds of getting three spins on a roulette wheel of 14-27-6 are *exactly* the same as spinning 32-32-32. We’d just call the first one “normal” and the second one “crazy,” because we see a pattern that we find interesting in one and not the other.
>if an event has a probability of occuring every 10 years, and it’s been more than 10 years since the last time it occured, we are living on borrowed time and the probability of it happening now is extremely high
This is a little different than the coin flip example, because a lot of real-world events aren’t totally independent, they’re dependent on a number of factors. Think of a volcano erupting for example – over time, magma will build up in a volcano, increasing pressure and increasing the odds of an eruption. There’s virtually a 0 chance of a volcano erupting a week after its last eruption, but a higher chance 5 year later, and a higher chance 10 years later (depending on the volcano, I guess), because the situation does change over time. Sometimes that can be the case for particular weather patterns or other events – they *may* become more likely over time. But it depends on the specific event, and what factors go into it – in some cases, the probability probably could get closer and closer to a near certainty over time.
Your friend is wrong.
After you have flipped the coin, the chance isn’t 0.5*0.5*0.5 (=0.125) anymore. It’s 1*1*0.5 (=0.5). Because the previous flips *have been* tails. So now their chance isn’t 0.5 anymore, but 1 because they undoubtedly did happen.
It’s like that fallacy in that joke about the guy who picked up a hitchhiker.
A guy picks up a hitchhiker. The hitchhiker asks “Why did you pick me up? I could have been a serial killer for all you know.” The driver says “Please. What are the chances that we’re both serial killers?”
It’s a fallacy, because the first guy is already a serial killer. It is a *set outcome* that he is a serial killer, so it has no influence on whether or not the other guy is a serial killer too.
Coin flips are independent events. The values of previous coin flips do not affect the values of future coin flips. Believing otherwise is known as the Gambler’s Fallacy.
Some events are not independent or are otherwise cyclical. In that case, knowing they have not happened for a long time can tell us something about the probability that they will happen now. Say I select my socks every morning at random and have one pair of purple socks. Each morning I select a non-purple pair of socks, the probability of selecting purple socks the next morning goes up (conversely, if I pick purple socks this morning, my chance of picking them next morning craters to 0). This doesn’t violate the Gambler’s Fallacy, because my sock selections are not independent – if I wear a pair one day, I can’t wear them some other day.
So the trick is to think about whether the events are independent or not. Most of the things we do in games of chance (flip coins, draw cards, generate random numbers, etc.) are independent. Many of the things we deal with in real life are not. The Gambler’s Fallacy mostly arises from us taking our real-world instincts into a special place (like a casino) where they don’t apply.
You are generally right. It depends on the circumstances of the circumstances of the problem.
If one event has an impact on the next even the next event, then his reasoning is good. An example of this would be drawing red balls out of a bag instead of yellow. There is a finite number of red balls in the bag, ao every time you draw another one out, the odds get higher against it happening again.
However if prior events have no impact on the next event, like your example of a coin flip, then the odds don’t change, so it would stay 50/50 as in your example.
If he still doesn’t understand it, ask him to explain to you how the previous coin flip physically impacts this coin flip. He won’t be able to do it.
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