Probability: Heads 5x in a Row - Illusion or Reality?

[malignant]

I mean I know what probability is, but let's say you flip a coin 5 times and it's heads every time...then switch coins? that can't have any affect on it but what if you waited an hour after getting 5 consecutive heads and come back with those 5 heads in mind, you would probably pick tails. but if you had forgotten about it instead of remembering it, then you would say it's 50/50 again. so which one is "right"? is it 50/50 again or is it in favor of tails? or is it just an illusion?

 

[HayleySarg]

I'm not entirely sure what you're asking.

Obviously variations in a coin may slightly alter the outcome(within the means of a rare-event rule case >5%) but the probability is still binary. It's either heads, or it's tails.

 

[malignant]

I'm not entirely sure what you're asking.

Obviously variations in a coin may slightly alter the outcome(within the means of a rare-event rule case >5%) but the probability is still binary. It's either heads, or it's tails.

I guess I'm asking if it ever "resets"? Like if you get 5 heads in a row, it's most likely going to be tails next but if you forget about the 5 heads then...yeah

 

[MarneMath]

If we're assuming the coin is fair then there should be no difference between the two different but fair coins (assume each coin has the same probability.) Since each flip of a coin is independent, then it doesn't matter if you use a new coin each time if they have the same probability.

 

[krash661]

it doesn't matter what you remembered,
it's still 50/50 because there's only two possible choices,
heads or tails.

getting heads 5 times in a row,
is still only a 50/50 choice.
it just means out of the 50/50 heads came up.

 

[HayleySarg]

Independent Events/Independence:

With less math theory:
http://www.mathgoodies.com/lessons/vol6/independent_events.html

With more theory:
https://en.wikipedia.org/wiki/Independence_(probability_theory )

 

[bp_psy]

I mean I know what probability is, but let's say you flip a coin 5 times and it's heads every time...then switch coins? that can't have any affect on it but what if you waited an hour after getting 5 consecutive heads and come back with those 5 heads in mind, you would probably pick tails. but if you had forgotten about it instead of remembering it, then you would say it's 50/50 again. so which one is "right"? is it 50/50 again or is it in favor of tails? or is it just an illusion?

You are talking about two different situations . The probability of getting six heads in a row is not the same thing as the probability of getting a sixth head given that you already had a five head sequence. In the second case the probability of having the first five head sequence is 1 (you already have it) so the probability of getting the six heads is 1*1/2 =1/2.
http://en.wikipedia.org/wiki/Conditional_probability

 

[malignant]

Oh I don't think I worded it right.

It's not exactly a mathematical question.

Say there's two sessions that you're going to flip coins, the second session is some unimportant time after the first. The main idea is that they're separate.

The first session you get 5 consecutive heads. If you had to bet money on the next flip, you'd pick tails because the probability of getting a 6th heads is 1/64. But if you decided not to flip the 6th coin until later on during the second session, keeping the 5 consecutive heads in mind, would you still bet money on tails?

During the second session, in your mind, you've still gotten 5 consecutive heads earlier during the first session so would you bet on tails or would you disregard it and pick one at random? If I was going to pick one at random anyways, then I would just pick tails because I have a slight reason to (the 5 consecutive heads earlier) even though it's always 50/50.

But in reality, what makes them separate and it not still being a 1/64 chance of getting a heads during the second session like it was after you got 5 heads in the first session? Basically what distinguishes 2 sets of probabilities at different times? Pure math doesn't really deal with time so that's why I said it's not really a mathematical question.

 

[malignant]

You are talking about two different situations . The probability of getting six heads in a row is not the same thing as the probability of getting a sixth head given that you already had a five head sequence. In the second case the probability of having the first five head sequence is 1 (you already have it) so the probability of getting the six heads is 1*1/2 =1/2.
http://en.wikipedia.org/wiki/Conditional_probability

that seems more related to what i was talking about. ill read up on it

 

[Office_Shredder]

The first session you get 5 consecutive heads. If you had to bet money on the next flip, you'd pick tails because the probability of getting a 6th heads is 1/64. But if you decided not to flip the 6th coin until later on during the second session, keeping the 5 consecutive heads in mind, would you still bet money on tails?

This is wrong, and you actually know it even if you don't realize it. Suppose I just start flipping a coin a bunch of times. After 4 or 5 minutes I happen to flip five heads in a row. I turn to you and say "I bet you my 1 dollar against your 60 dollars that it comes up heads on the next flip"

If you think the odds of the next flip coming up heads is 1/64, then you should think my bet is a fair bet (even if you don't take it because most people don't like betting money on even odds). But most likely you're thinking that the bet sounds outrageous, because there's a 50/50 chance that the next flip is a heads vs a tails

 

[bp_psy]

The first session you get 5 consecutive heads. If you had to bet money on the next flip, you'd pick tails because the probability of getting a 6th heads is 1/64. But if you decided not to flip the 6th coin until later on during the second session, keeping the 5 consecutive heads in mind, would you still bet money on tails?

No the probability is not 1/64 it is 1/2 since you already had 5 heads in a row. At this point the first 5 head sequence event is certain that is p_5H=1. The entire outcome is determined by the next flip since the previous 5 heads are given. If you had 1000 head sequence your odds of getting 1001 are still 1/2 provided that your coin is fair.If you bet on the next flip the odds are always 1/2. On the other hand if you had to bet at the beginning for a sequence of n heads the odds would be much smaller then 1/2.

 

[MathJakob]

You're misunderstanding. It doesn't matter if the coin lands on heads 100 times in a row, the next flip is still 50/50 to land on heads again no matter if you flip immediately or leave it for a day or even use a different coin.

Before I start flipping the coin if you asked me what is the propability to get heads 12 times in a row for example, then it would be 1/2048. But each individual flip has equal chance.

 

[malignant]

You're misunderstanding. It doesn't matter if the coin lands on heads 100 times in a row, the next flip is still 50/50 to land on heads again no matter if you flip immediately or leave it for a day or even use a different coin.

Before I start flipping the coin if you asked me what is the propability to get heads 12 times in a row for example, then it would be 1/2048. But each individual flip has equal chance.

i see now. only thing that's weird to me is how the outcomes always tend to even out right? like 1000 flips tends toward 500/500 so that's why i was thinking if it was say 999 heads then the final flip would more likely be tails. misleading on my intuition

 

[HayleySarg]

Ah, so the law of large numbers then?

http://en.wikipedia.org/wiki/Law_of_large_numbers

The more trials you do, the closer the results approach the expected result.

 

[MathJakob]

i see now. only thing that's weird to me is how the outcomes always tend to even out right? like 1000 flips tends toward 500/500 so that's why i was thinking if it was say 999 heads then the final flip would more likely be tails. misleading on my intuition

Psychologically you'd probably pick tails just because you'd say to yourself surely it can't be heads again... but it really makes no odds.

It's the same reason why people who play the lottery never pick 1 2 3 4 5 6 7 just because psychologically it is less likely the occur than 7 random numbers, even though it has exactly the same probability to occur than any other set of 7 numbers.

 

[zoobyshoe]

Derren Brown did a coin flip thing. He bets the viewer he can get ten heads in a row. Then they show a continuous shot of him flipping a coin, and he gets ten heads in a row.

The 'trick' we find out, is that the clip shown was the final minute out of something like ten grueling hours of coin flipping. After all those hours, he finally, accidentally, got ten heads in a row.

I don't know what that says about probability.

 

[xxChrisxx]

Say there's two sessions that you're going to flip coins, the second session is some unimportant time after the first. The main idea is that they're separate.

During the second session, in your mind, you've still gotten 5 consecutive heads earlier during the first session so would you bet on tails or would you disregard it and pick one at random? If I was going to pick one at random anyways, then I would just pick tails because I have a slight reason to (the 5 consecutive heads earlier) even though it's always 50/50.

So you are saying that your brain is subject to gamblers fallacy. The correct answer is you'd just pick one at random.

The more smart *** answer is to bet on heads, with a disproportionate amount of heads over a sample size, the more likely the coin is biased.

 

[256bits]

i see now. only thing that's weird to me is how the outcomes always tend to even out right? like 1000 flips tends toward 500/500 so that's why i was thinking if it was say 999 heads then the final flip would more likely be tails. misleading on my intuition

Well not exactly, that is not quite correct. For a a coin toss, as the number of tosses increases the percentage of heads or tails tends towards the mean of 50%.. The actual number of heads and tails will diverge from the mean of equal number of heads and tails.

 

[MathJakob]

Well not exactly, that is not quite correct. For a a coin toss, as the number of tosses increases the percentage of heads or tails tends towards the mean of 50%.. The actual number of heads and tails will diverge from the mean of equal number of heads and tails.

If you made a computer that picked a number from 1 - 10 after an hour of picking millions of numbers you'll see that 1,2,3,4,5,6,7,8,9 and 10 are all picked the same amount of times, well almost the same and the longer the program runs the closer the averages get to being equal.

Isn't this just the law of averages?

 

[malignant]

So you are saying that your brain is subject to gamblers fallacy. The correct answer is you'd just pick one at random.

The more smart *** answer is to bet on heads, with a disproportionate amount of heads over a sample size, the more likely the coin is biased.

no, that's not what i was referring to but my post had errors. i meant according to the law of large numbers or law of averages or whatever you want to call it. so it wasn't really a gamblers fallacy. i see now that the next flip is still 50/50 but i meant more as a whole.

 

[256bits]

If you made a computer that picked a number from 1 - 10 after an hour of picking millions of numbers you'll see that 1,2,3,4,5,6,7,8,9 and 10 are all picked the same amount of times, well almost the same and the longer the program runs the closer the averages get to being equal.

Isn't this just the law of averages?

The probability of picking any number from 1 to 10 is 0.1. With the law of averages, as more picks are performed the frequency of each digit picked will approach 0,1. Setting up a histogram of number of picks of each number versus its frequency will show this quite well, with the top becoming more flat as the trial progresses. The law of averages expects each number to turn up 100,000 times, on average.

On the other hand. a plot of the absolute number of picks of each number reveals a histogram becoming more jagged as the number of picks increases. An excess of say the digit 3 coming up 800 times more than any other with a milliion picks still falls close to the expected frequency of 0.1 ( some of the other numbers had to be picked less than the 100,000 expected ). An excess of 1300 with 2 million picks falls even closer.

What is going on here has to do with the law of averages and the the law of large numbers.

Which kind of explains the following. If you toss a coin 10 times and 10 heads show up, one would immediately conclude that something extraordinary is occurring, and that the coin is biased, or that due to the law of averages the next toss with turn up a tails, and that subsequent tosses will turn up more tails than heads to even things out. With many many many throws, though, the run of 10 heads has just as much chance of occurring starting from the first throw to the 10th from last throw. If it starts happenning at the 384,269 throw then one would not think much of it. If your friend comes in at the beginning of the 384,269 tos and sees the 10 head run, he on the other hand might be mystified and think what are the chances of that ever happening.
Fact is one does not know where in the sequence of tosses one is starting his own personnel count from, and as each toss is an independent event with the coin not caring what its previous toss was or what other coins are doing, it really does not matter.

 

[micromass]

no, that's not what i was referring to but my post had errors. i meant according to the law of large numbers or law of averages or whatever you want to call it. so it wasn't really a gamblers fallacy. i see now that the next flip is still 50/50 but i meant more as a whole.

The strong law of large numbers (SLLN) does say that if you toss a coin n times and if n is very large, then close to n/2 tosses will be head and close to n/2 tosses will be tail.
The problem with this is that the SLLN does not give any information about what "very large" is and what "close to" is. This is why you can't infer anything from it with respect to one coin toss. Let's say you toss 1000 times and you have 600 head and 400 tail. You can't infer anything about the next toss being tail more likely. because you don't know whether 1000 tosses is considered to be "very large" for the SLLN, or whether a distance of 100 is considered to be "close to".

 

[micromass]

Here's another thing to consider.
Let's say that we throw a fair coin a 1000 times and we get all heads. Must we then somehow get more tails than heads in what comes next? No, the SLLN does not imply such a thing.
Indeed, let's say that the next throws are all head, tail, head, tail, head, tail, ... So we throw as many heads as tails. So we have in total thrown more heads than tails. This does not contradict the SLLN. The averages will still converge to 1/2 head and 1/2 coins.

The entire point is that a finite number of anomalies are not important for the SLLN. If we want the average to change, then we need something to happen an infinite number of times.

 

[dipole]

People seem to be going in circles with words here...

Let's use three coin flips. Suppose I flip a single coin three times, obviously the probability of getting heads for each coin flip is 1/2, but let's enumerate all the possible outcomes of the three flips, we have:

T T T
T T H
T H T
H T T
T H H
H T H
H H T
H H H

Out of eight possible set of outcomes, only one is all three with heads in a row. The probability of flipping a coin three times and getting all heads is (1/2)(1/2)(1/2) = 1/8. However, the probability for getting heads for each single flip is always 1/2.

If I ask the question "what is the probability I will get heads four times in a row if I flip a single coin four times" then the answer is 1/16, because on average you'd only get that outcome once for every sixteen times you tried. If you ask the question "what is the probability that, after three coin flips, the fourth coin flip is heads?", then the answer is 1/2.

They are distinct questions.

edit: Note that this means that suppose I took a huge number of coins and laid them out in an nx3 grid, and used this to simulate flipping the same coin 3n times. If I collected all the rows that had all heads, and then added a fourth coin to each row, with the side facing up chosen at random, then on average half of this new set of rows will be H H H H and the other half will be H H H T.