ELIF I don’t get probability. How come the probability of getting heads in a coin toss is still 50/50 even after you have had tails for straight five times a row. Shouldn’t the probability of getting tails six times in a row be lesser than getting a head this time?
In: Mathematics
Everything being well explained here. Just want to add this for the gambler’s fallacy:
Monte Carlo Casino
Perhaps the most famous example of the gambler’s fallacy occurred in a game of roulette at the Monte Carlo Casino on August 18, 1913, when the ball fell in black 26 times in a row. This was an extremely uncommon occurrence: the probability of a sequence of either red or black occurring 26 times in a row is (18/37)26-1 or around 1 in 66.6 million, assuming the mechanism is unbiased. **Gamblers lost millions of francs betting against black, reasoning incorrectly that the streak was causing an imbalance in the randomness of the wheel, and that it had to be followed by a long streak of red.**
from: [https://en.wikipedia.org/wiki/Gambler%27s_fallacy#Monte_Carlo_Casino](https://en.wikipedia.org/wiki/Gambler%27s_fallacy#Monte_Carlo_Casino)
Thank you people! I realize that I wasn’t smart enough to understand the idea of “dependent” and “independent” events. Also, probability and luck are different things.
Your assumption is called the gambler’s fallacy.
If you flip a fair coin 99 times and get 99 tails, your next flip is 50/50 heads or tails. just because something rare happened before does not mean something rare will happen.
Both of these things are true at once:
* The chance of getting Tails 6 times in a row is only 1.6%.
* If you get Tails 5 times in a row, then on the next toss, the chance of getting Tails is 50%.
The 50% chance is the chance to get Tails on any 1 toss. The first toss is a 50/50 chance; the 6th toss is a 50/50 chance; the millionth toss is a 50/50 chance. This is because each toss has **nothing to do with the previous tosses** – each one is its own 50/50 chance. These are called “independent events”.
The chance of getting Tails 6 times in a row **depends** on the 50/50 chance for toss 1 being Tails, *and* the 50/50 chance for toss 2 being Tails, *and* the 50/50 chance for toss 3 being Tails, etc. That 1% chance is the total chance for the **dependent event** of “6 Tails in a row”. But, note that each individual one of those tosses was still 50/50 by itself.
Let’s say I pull a coin out of my pocket and hand it to you. I tell you I’ve just flipped it 100 times and gotten either all heads or all tails. Is there any way for you to examine the coin and tell me which way the previous coin flips went? What if I’m lying? Can you tell that by examining the coin?
Unless you’re uncovering the fact that there is something “wrong” with the coin that makes it more likely to come up heads (or tails), there’s nothing insightful about a string of heads (or tails) in multiple flips. Each new flip happens completely independently of all prior flips.
Instead of imagining 10 flips in a row, imagine 10 people flipping 10 identical coins at the same time. The odds of all 10 coming up heads on the same flip are incredibly small, but repeat it enough times and it will happen. When it does you wouldn’t think any given coin impacted any of the other coins. You’d chalk it up to chance. Same with one coin coming up heads 10 times in a row.
Divide the number of events by the number of possible outcomes. This will give us the probability of a single event occurring. In the case of rolling a 3 on a die, the number of events is 1 (there’s only a single 3 on each die), and the number of outcomes is 6, so a 1/6 chance.
In the case of a coin, each individual instance has a 50% chance of landing either heads or tails: 1 event over 2 outcomes. If you are looking at a string of 6 tails in a row, you take 1 event (all tails) over the number of possible outcomes, 64, to get a 1/64 chance (~1.5%)
Each coin flip is an “independent” event, so it’s unaffected by what happened on the previous flip or the next flip. The reason you feel like it should be less after a bunch of tails in a row is that you have an intuition that the probability of that same thing happening a lot of times in a row is very low. Each time you flip a coin, the probability of getting tails is 0.5, but the probability of getting tails 6 times in a row is 0.5 * 0.5 * 0.5 * 0.5 * 0.5 * 0.5.
Think of it this way – the coin doesn’t “know” what was thrown the past 5 times. It only knows it had two options here, heads or tails, both equally likely.
The likelihood of getting tails 6 times in a row is lower than 50:50 because on *each* throw you’re getting a 50% chance at heads. So for all those throws to align is pretty unlikely. But the coin doesn’t “know” that. It just knows it’s two options – heads or tails; 50:50 chance.
The 6 tails in a row is sort of a meta probability compared to the individual coin tosses.
The reason why it’s still 50/50 is because coin tosses are time invariant. Results from current coin tosses are not dependent on coin tosses from the past.
As for your example, if you get tails six times in a row and you get tails again, that is just as likely as getting heads after getting tails six times. If you have any other questions, do ask
In ELI5 terms: the coin has no memory like you do, so it will stay 50/50
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