#Probability #CoinFlip #GamblerFallacy #Statistics
Have you ever wondered about the probability of flipping 5 tails in a row and whether it affects the chances of flipping a tail in the next flip? 🤔 Let’s dive into the world of probability and statistics to understand why flipping 5 tails in a row still results in a 50/50 chance of getting a tail in the next flip.
Understanding Probability in Coin Flips
When it comes to flipping a coin, there are two possible outcomes: heads or tails. Each outcome has an equal probability of 1/2 or 50%. This means that every time you flip a coin, the likelihood of getting a head or a tail is the same.
The Gambler’s Fallacy
The gambler’s fallacy is the mistaken belief that if an event occurs more frequently than expected in the past, it is less likely to happen in the future. In the context of flipping a coin, this fallacy leads people to believe that after a series of tails, a head becomes more likely to occur. This is a misconception because each coin flip is an independent event and is not influenced by previous outcomes.
Explaining Independent Events
When you flip a coin, the outcome of each flip is completely independent of previous flips. Regardless of how many tails you have flipped in a row, the probability of the next flip resulting in a tail remains 1/2. Here’s why:
Each coin flip has its own separate probability, and the outcome of one flip does not affect the outcome of another flip.
The coin doesn’t have a memory, so it doesn’t “know” how many tails have been flipped before.
Even if you have flipped 5 tails in a row, the next flip still has a 50% chance of being a tail.
To put it simply, the outcome of each flip is determined by chance and is not influenced by past outcomes. The coin doesn’t have a bias towards one outcome just because the previous flips resulted in the same outcome.
Visualizing Probability
Consider the example of rolling a six-sided die. Each time you roll the die, there is a 1/6 chance of getting a specific number. If you roll the number 4 three times in a row, the probability of rolling a 4 on the next roll is still 1/6. The previous outcomes do not change the likelihood of rolling a 4 in the future because each roll is an independent event.
Resisting the Fallacy
It’s important to recognize the gambler’s fallacy and understand that past outcomes do not affect future probabilities. The belief that after a series of tails, a head is “due” is a misconception that can lead to poor decision-making, especially in gambling situations. By understanding the principles of probability and independence, you can make more informed decisions and avoid falling into the trap of the gambler’s fallacy.
Final Thoughts
In conclusion, the probability of flipping 5 tails in a row does not impact the chances of getting a tail in the next flip. Each coin flip is an independent event with a 50% probability of resulting in a tail. The gambler’s fallacy arises from the misconception that past outcomes influence future probabilities, but in reality, each flip is determined by chance and is not affected by previous flips.
By understanding and embracing the principles of probability and independence, you can make more informed decisions and avoid falling into the trap of the gambler’s fallacy. So, the next time you flip a coin, remember that each flip is a fresh start with a 50/50 chance of landing on heads or tails. Stay curious and keep exploring the fascinating world of probability! 🎲🤓
Before you start flipping the coin, flipping 6 tails in a row is indeed pretty unlikely. However, by the time you’ve flipped it 5 times, the “unlikely” part has already happened. Taken as a whole, the sequence is unlikely, but the fallacy comes from the fact that at the 5th flip you’re not dealing with the whole sequence anymore, but just that single 50-50.
Think of it another way: if I showed you two (perfectly ordinary) coins and told you that one of them just flipped 6 heads in a row, what could you possibly do to determine which one of them is more likely to land tails on the next flip? None of the flips you do change the coin in any way to make one outcome more or less likely.
The basic issue is that the coin has no memory. It *cannot* know what happened in the previous flips. Nor can any other part of the system. Since there is no memory of the past, there can be no way for the past to influence the future. So the next flip is 50/50, just like all the others.
The coin doesn’t have a conscience. It doesn’t know how many times it landed head or tail before. Therefore it just does what physics dictates it, and that is landing on either with a 50 % chance.
You said it yourself. Each coin flip is independent. The coin doesn’t know about what happened before, it will always give you a 50/50 chance for either heads or tails.
Think about it this way. If you were to toss a coin five times and you get heads every time, then you could give that coin to another person and ask them to toss it for you. It will still be a fair 50/50 for heads or tails. They don’t know the previous result, why should the probability be any different for them?
Because each flip is independet of each other. A series of flips depends on one another, so saying getting 6 tails in row is not 50/50. But every flip by itself has a 50/50 chance.
Let’s simplify this a bit and flip 3 coins in a row instead of 6 coins in a row.
There are 8 different possible outcomes (H for heads, T for tails):
HHH HHT HTH HTT THH THT TTH TTT
So let’s say I flipped 2 tails in a row. Which means that after my next coin flip, the results are going to be either TTH or TTT. Either one of these sequences has a 1 in 8 chance of happening when flipping a coin 3 times in a row. In other words, you are exactly as likely to flip 2 tails in a row followed by a heads, as you are 3 tails in a row. That’s why the last coin flip is still a 50/50 chance.
So going back to your example, the chance of flipping 6 tails in a row is 1/64, but the chance of flipping 5 tails in a row followed by a heads is also 1/64.
independent random events every flip has the same 50/50 odds. this is the same concept slot machines use
with a large enough sample size it will always end up with 50 50 but in a smaller sample size that may not show up
you mention flipping 6 in row and flipping once the two are completely separate from each other
Provided it’s a fair coin, each flip is an independent event with a probability of 50% heads, 50% tails, you are correct. The coin has no memory; it doesn’t care what happened in the previous flips.
This does mean that over a large number of trials (flips) you would expect approximately the same number of heads and tails.
This does create something of a logical paradox because, if heads is “winning” 5-0 and over time you expect equal numbers, the probability of tails appears to have to increase.
It doesn’t; it remains 50/50
The math behind what you’re referring to is called conditional probability. So the probability “A” (that you flip 6 tails in a row) given that “B” (you already flipped 5 tails in a row) is Prob(A) / Prob(B) = 1/64 ÷ 1/32 = 1/2.
But conceptually, you just have to realize that flipping a coin is an individual event. Think of it this way… If you roll a 6 sided die, the probability of rolling an even number is also 1/2. So your odds of rolling an even number on a die and then flipping a tails on a coin is 1/4. Let’s say you roll a die and get an even number. So now that you’re about to flip the coin, do you think the roll of your die will affect the flip of your coin? Would your odds of a tails be different if you had rolled an odd number?
>it feels right (as fallacies often do) that in consecutive flips the previous events matter?
Why does it feel right?
How would a coin know what was the previous flip?
What would the mechanism be?
Do you believe in some force like luck or fate or God controlling what happens with the next coin flip?
The gamblers fallacy isn’t a true “fallacy” at all. It’s a legitimate heuristic that sometimes gets applied incorrectly.
In any case, the heuristic you’re using is correctly applied to something like a shuffled deck of cards where a card is drawn without random replacement and one asks “what is the probability of drawing the ace of spades?” Each time you draw a new card the deck *physically changes*. It goes from 52 cards to 51, to 50, and so on. The odds of drawing the ace of spades goes up with each trial because the trial changes the state of the deck. The heuristic is also correctly applied widely to things like earthquakes (the probability of an earthquake tomorrow goes up each day as the tension between plates increases), Russian roulette, and searching for lost items (as you search each room in the houses the number of possible places the missing item could be goes down and, therefore, your odds of finding it in the next room goes up).
The heuristic is misapplied with things like flipping coins) because **the act of flipping a coin does not change the physical characteristics of the coin**. Other misapplications include playing slot machines, buying lottery tickets, drawing from a deck of cards *with* replacement, etc.
So when you are wondering when and where to apply this heuristic, ask yourself if each trial or event changes the state of the system. And if it does change the system ask yourself *how* it changes the system. There is no rule here to follow other than to build a mental model of the system you’re talking about.
Assuming it’s a fair coin, it doesn’t know what happened in the earlier flips. Once you pick up the coin to flip again, you’ve removed the previous state, so the last flip can’t possibly influence the next flip. You seem to know this consciously, but it feels wrong to you because of the five in a row.
The thing about randomness is that it’s random, but your brain isn’t always good at knowing whether data looks random. If a coin strictly alternated heads and tails, that wouldn’t be random. So you have to learn to expect a certain amount of clustering. Two or three in a row should happen very frequently, and five or six in a row should happen occasionally. Even some short alternation sequences should happen from time to time. That’s part of the randomness, and your brain will see little patterns that aren’t there, and will expect them to continue.
The other thing specifically about getting multiple consecutive flips the same is that you begin to suspect the coin isn’t perfectly random. It does become more likely at some point that the coin is biased or even tails on both sides, and in those cases it would be rational to expect the pattern to continue. But that’s not randomness anymore.
I think you grasp the concept that each flip is independent well enough, but a thing to remember is that every distinctive series has the same chance of happening as any other, so HHHHHH is as (un)likely to happen as HTHTHT.
The chances of eventually getting 3 H and 3 T are higher because there are more routes to get that total, but only one route to get 6 H.
It’s just that each flip eliminates all the other routes that had a chance at the beginning.
If you flipped a coin thousands of time and looked at every sequence that had 5 tails in a row than the sixth flip in that sequence would be fairly evenly divided between heads and tails.
I would like you to explain to me why you think this feels right? You clearly understand why the 50/50 chance is accurate. You explained it in your own post. To me it feels very wrong that the odds would be anything but 50/50, it would change my entire view of how the world works if that were not the case.
Each flip is its own completely unique event, and no other flip has anything to do with it, so it’s always 50/50.
The coin doesn’t know that the last flip was heads. Each event is its own event, with no link to previous or future events.
Let us pretend that a machine flipped a coin 6 times and it recorded each result.
HTHTHT would be heads-tails-head-tails-heads-tails
HHTTHT would be heads-heads-tails-tails-heads-tails
Now this machine does this 6 flip and record exercise one million times, and you have access to that table.
If you go look for how many of the entries are TTTTTT (six tails) you will find that it is 1/64 of the total number of recorded entries.
Next you go look for every entry that starts with five tails, to TTTTT* where the * is either H or T. If you put all of those entries together you will find that all of them are 2/64 of the total of all recorded entries. And if you look at them the number of TTTTTT entries and TTTTTH entries are the same.
From an emotional point of view, it does feel like the heads is “due” to come up. But if it helps think of it this way “what are the chances of me flipping 6 tails in a row?” is a different question than “what is the chance of me flipping a tail, given that there were 5 flips before it?”
The odds of flipping 4 tails and then a head is 1/64. The odds of flipping 4 tails and then another tail is 1/64. So, once you’ve flipped the 4 heads, odds are even either way.
But, that assumes a “fair coin.”. After, say, 100 tails in a row, you might start thinking “this isn’t actually a fair coin.”.
Each flip of a single coin is an independent event with a 50:50 chance for each option.
So, while the odds of flipping a coin, and having it come up “heads” 6X in a row is 1/64, each INDIVIDUAL flip is still a 1/2.
Streaks are normal in a something that is random. If you want to see a fun example of this, check this [numberphile video](https://www.youtube.com/watch?v=tP-Ipsat90c)
The odds of flipping XXX XXX or OOO OOO…
… are exactly the same as you flipping XXX XXO or XOX OXO.
Each individual flip is 50-50, but the odds of achieving ANY specific pattern of 6 is 1/64.
Simple answer: flipping TTTTTT has a 1/64 chance, but so does flipping TTTTTH. They have the exact same probability. So when you already got TTTTT those are the only two options left and they have the same probability.
>I know this is the gambler’s fallacy, but why is it a fallacy?
Because each flip is a discrete event, and the prior events have no impact on the future events. There’s no law or mechanism of the universe that’ll reach out and say “okay, that’s enough tails, now you get heads.”
The coin doesn’t care what happened before. Each time you flip it, there are only two options: head or tails.
Consider the following 6 flip sequences:
h-h-t-t-h-h
t-h-h-h-t-h
h-h-h-t-t-t
h-t-h-t-h-t
t-t-t-t-t-h
t-t-t-t-t-t
Every single one of them has the exact same probability of occurring, 1/(2^6) = 1/64. Every single coin flip sequence of n flips has a 1/(2^n) probability of occurring if you flip a coin n times.
If you want to, you can take the time to write out all 64 possible unique sequences of 6 coin flips. Since there are 64 sequences and one of them is t-t-t-t-t-h, this sequence has a 1/64 chance. Same for t-t-t-t-t-t.
Let’s say you flip five tails in a row and then drop the coin on the ground and lose it. A week later, somebody finds the coin and flips it. What are their odds of getting tails? 50/50, right? Why? Because the coin doesn’t store luck. It doesn’t store luck over the course of a week and a change of ownership and it doesn’t store luck over the couple of seconds that it takes to flip it again. They’re independent events.
Here’s where I will always disagree with the “standard” answer, at least as far as a larger number of flips (say 100) goes. If I flip a coin 99 times and it lands on tails 99 times in a row, the odds against that happening are astronomical! This would lead me to conclude there is another factor at play. I would question whether the coin is weighted differently, or if there is some sort of strange magnetism or environmental condition happening, or even a strange gypsy curse on the coin, it doesn’t matter. It would be so phenomenal to have 99 flips land on tails, it is actually **more** reasonable to assume that whatever phenomenon has caused this will also cause it to land on tails again.