I know, 50/50 heads tails right? But help me understand the next step – each coin flip has a 50/50 shot of heads or tails. What I don’t understand is how the likelihood of the next flip doesn’t change. For example if I flipped a coin 10 times and every time it flipped heads, the next flip would be 50/50 tails. Wouldn’t the likelihood of flipping a coin 11 times and having it be heads every time be really low? 0.5^11 = 0.048%?
Here’s the origin of the question. I was at a roulette table and the guy said “it’s been black the last 8 rolls, the next one has to be red.” At first I thought, the next roll will be ~47% black, ~47 red, ~6% 0 or 00 you fucking imbecile. Then I thought to myself, what are the chances that there are no red rolls in 9 rolls, which is well below 1%.
Am I the imbecile?
In: 215
The usual model would say that 8 blacks in roulette do not change the probabilities on the next roll. LIkewise with coin tossing. For example, in two coin tosses, with the usual model, the following outcomes have equal probability: HH, HT, TH, TT. Getting a heads does not change the probability of getting heads on the next toss. Whether or not the standard models do a proper job of modelling reality is another question.
Please take a look at [this video](https://youtu.be/tP-Ipsat90c). Maybe it will make more clear for you. This yt channel have more videos on this topic so maybe you can check them out, these guys are great at explaining things.
You’re conflating a single event with a series of events.
When looking at a individual event, yes…. the NEXT flip has a 50/50 chance. It doesn’t matter what comes before. They are not related in any way.
A specific series of events that lead up that that flip might be less likely.
To expand, let’s say we are going to flip 3 total times.
Each individual flip has a 50/50 chance.
But with 3 flips, there are 8 total potential sequences… each *sequence* has a 12.5% chance of occurring.
H H H
H H T
H T H
H T T
T H H
T H T
T T H
T T T
So, yeah, you have a 12.5% chance of getting 3 heads in a row…. but you ALSO have a 12.5% chance of gettting two heads in a row followed by a tails. and ALSO have a 12.5% chance of getting Tails, Heads, Tails. Each sequence has the same likelihood of happening.
… and you can think of a ‘single flip’ as a ‘sequence’ of one… where there are only two options – heads or tails.
A lot of times, with probabilities, it helps to literally write out all the options… then you start seeing a pattern and can extrapolate to bigger numbers.
Think of it this way: Let’s say that last week you were unlucky enough to be hit by lightning, then developed a rare disease. That doesn’t mean you can drive without a seatbelt on the way home thinking “Well, nothing else bad can happen to me.” Your chances of a car accident have nothing to do with whether you have the disease or were struck by lightning.
Now, two weeks ago, we would have said “The chances of you getting hit by lightning, developing a rare disease AND being in a car accident in a two week period is really really small.” But, once the first two of those happen, the accident doesn’t become any less (or more) likely to to happen.
The best way to think about this (IMHO) is as follows:
The chance of getting heads (or tails) from a fair coin is 50% or 1 in 2. This means that, if we flip a coin a lot of times, chances are (close to) half of the flips will be heads, and (close to) half will be tails.
When we say the chances of 4 heads in a row is 1/16 (1/2 for the first, 1/4 for 2 in a row, 1/8 for 3 in a row, and 1/16 in a row), what we are saying is: if we flip a coin ___4 times a lot of times___, that is to say, we perform a series of 4 flips over and over, (close to) 1/16 of those series will be all heads.
Each individual coin flip is ignorant of every other, and whatever happened before doesn’t matter: the next coin always has a 50:50 chance of being heads or tails.
The odds of an event are altered for things that have already happened. What are the odds that the last eight roulette spins were black? If you weren’t watching, you might say they were less than 1%. If you did watch the last eight spins and they *were* all black, it doesn’t matter what the chances are normally: the chance of it having happened changes to 100%.
Let’s say you enter a lottery where your chance of winning is one in a million, and then you toss a coin where the chance of you winning is one in two.
The odds of you winning the lottery *and* winning the coin toss is one in two million.
But: if you already won the lottery, the chances of you completing the “win the lottery then win a coin toss” game changes, because we’ve established that you did the hard part already. So at that point the odds of you winning the lottery and winning the coin toss are now one in two.
I’m reminded of the Martingale betting system. It basically states you bet one unit (for our sake let’s just say our bet is $1), and then double it when you lose.
In this way if you bet and win, you are up $1. If you bet and lose then double your bet to $2. If you lose again, double it to $4. When you win, In this case $4, you are actually only up your original bet. You won $4, but lost $1 and $2. So net $4.
So if this was your betting system, you could analyze it and determine that if I had $63, which is enough for 6 bets ( 1,2,4,8,16,32) what is your chance of total failure. So 6 lost bets in a row. Well for roulette let’s pretend betting black is 50%. It isn’t as you know because of 0 and 00 but I want easy math.
6 losses in a row would have a probability of 0.5^6 ~ 1.6%. So my odds of success would be about 98.4%, of ending up ahead $1.
That’s the expectation. BUT when I’m at the casino, and I’m that 6th bet, with my $32 down on black, my odds are only 50% for this result not 98%. We are here now. There was a 50% for each spin to come up black, but it hasn’t. But each spin had the same odds.
Something that helped me with thinking about this is that *every* configuration has very low odds.
TTTTTTTTTT has odds of 1/1024.
HHHHHHHHHH has odds of 1/1024.
So do:
THHTHHTHTT
HTTTHTHTTH
TTTTHTHHHH
The disconnect comes from us seeing HHHHHHHHHH as a special result, and HTHTTHHHTT as unremarkable, when they’re both equally unlikely.
50/50 just means that half the tries are being one result.
But of how many tries?
It can be 5 of 10, 500 of 1000 or even 500m of 1b.
There was once a case in 1913 in the Monte Carlo Casino.
There was a roulette game going on with the ball falling 26 times in a row on black, even though the chances are ~50%.
The chance of each single toss doesn’t change
The chance of heads = 1/2
The chance of two heads in a row = 1/4
If you have already tossed it once and it was heads, then instead of 4 possibilities for 2 heads in a row, there is only a 1/2 chance, heads or tails. The Tails/heads and the Tails/Tails options don’t apply anymore, you have already eliminated half of the possibilities.
If you toss it 9 times in a row and they are all heads, the chances the 10th one will be heads is still 1/2
The chances in general of tossing 10 heads in a row are very small, but you are already down that path, so the chances of what’s in front of you is still 1/2
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