How do independent events not affect each others probability of occurring?
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I am severely confused with regards to the idea of a succession of independent events and how each event does not affect the probability of the other.
For example: Assume a fair coin has been flipped 9 times resulting in all heads. Each coin flip has no physical affect on the next resulting in an independent probability. Thus, the chance of flipping heads on the next flip has a 1/2 chance of happening. However, according to a proposition in my textbook, the chance of flipping 10 heads in a row is 1/2^10 = 1/1024.
This implies a contradiction which means at least one of my implied or explicit assumptions is wrong.
Where am I wrong? I would greatly appreciate any sort of rigorous and or intuitive explanation.
Thanks
probability probability-theory
asked Nov 22 at 13:31
Kohler Fryer
1031
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I am severely confused with regards to the idea of a succession of independent events and how each event does not affect the probability of the other.
For example: Assume a fair coin has been flipped 9 times resulting in all heads. Each coin flip has no physical affect on the next resulting in an independent probability. Thus, the chance of flipping heads on the next flip has a 1/2 chance of happening. However, according to a proposition in my textbook, the chance of flipping 10 heads in a row is 1/2^10 = 1/1024.
This implies a contradiction which means at least one of my implied or explicit assumptions is wrong.
Where am I wrong? I would greatly appreciate any sort of rigorous and or intuitive explanation.
Thanks
probability probability-theory
asked Nov 22 at 13:31
Kohler Fryer
1031
The chance of flipping 10 heads in a row is very unlikely don't you think? The chance of flipping any string of 10 heads or tails is $frac{1}{2^{10}}$ if you specify that string in advance.That is a completely different question from the chance of getting a heads on one throw (regardless of what you flipped before or the weather or anything else). There is no contradiction here.
– Paul
Nov 22 at 14:01
So given that I flipped 9 heads in a row, the chance that I flip 10 heads in a row is 1/2? And the chance of flipping 10 heads in a row is 1/1024?
– Kohler Fryer
Nov 22 at 14:20
1
You’re missing that it’s already highly unlikely that you’ve flipped nine heads in a row.
– Randall
Nov 22 at 14:28
1
Let's go frequentist here. You sit down and flip ten coins and record the pattern of H and T. You are very unlikely to get HHHHHHHHHH don't you think? Do the experiment again, and again and again. On average, you will only get HHHHHHHHHH every 1024 attempts. Now think about your friend flipping the same coin 9 times but you can't see what he got. You throw the coin once to make it 10 times - how likely is a head on your coin?
– Paul
Nov 22 at 14:34
1
Following Paul's second comment, suppose you've done the 10-flip experiment a huge number of times, say a billion. Out of those billion runs, only about 1 in 1024 will be all heads. About 1 in 512 will begin with 9 heads. Of those that begin with 9 heads, about half will continue with a tenth head. That's what conditional probability 1/2 means.
– Andreas Blass
Nov 22 at 18:49
|
show 1 more comment
up vote
0
down vote
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up vote
0
down vote
favorite
I am severely confused with regards to the idea of a succession of independent events and how each event does not affect the probability of the other.
For example: Assume a fair coin has been flipped 9 times resulting in all heads. Each coin flip has no physical affect on the next resulting in an independent probability. Thus, the chance of flipping heads on the next flip has a 1/2 chance of happening. However, according to a proposition in my textbook, the chance of flipping 10 heads in a row is 1/2^10 = 1/1024.
This implies a contradiction which means at least one of my implied or explicit assumptions is wrong.
Where am I wrong? I would greatly appreciate any sort of rigorous and or intuitive explanation.
Thanks
probability probability-theory
asked Nov 22 at 13:31
Kohler Fryer
1031
I am severely confused with regards to the idea of a succession of independent events and how each event does not affect the probability of the other.
For example: Assume a fair coin has been flipped 9 times resulting in all heads. Each coin flip has no physical affect on the next resulting in an independent probability. Thus, the chance of flipping heads on the next flip has a 1/2 chance of happening. However, according to a proposition in my textbook, the chance of flipping 10 heads in a row is 1/2^10 = 1/1024.
This implies a contradiction which means at least one of my implied or explicit assumptions is wrong.
Where am I wrong? I would greatly appreciate any sort of rigorous and or intuitive explanation.
Thanks
probability probability-theory
probability probability-theory
asked Nov 22 at 13:31
Kohler Fryer
1031
asked Nov 22 at 13:31
Kohler Fryer
1031
asked Nov 22 at 13:31
Kohler Fryer
1031
asked Nov 22 at 13:31
Kohler Fryer
1031
asked Nov 22 at 13:31
Kohler Fryer
1031
1031
The chance of flipping 10 heads in a row is very unlikely don't you think? The chance of flipping any string of 10 heads or tails is $frac{1}{2^{10}}$ if you specify that string in advance.That is a completely different question from the chance of getting a heads on one throw (regardless of what you flipped before or the weather or anything else). There is no contradiction here.
– Paul
Nov 22 at 14:01
So given that I flipped 9 heads in a row, the chance that I flip 10 heads in a row is 1/2? And the chance of flipping 10 heads in a row is 1/1024?
– Kohler Fryer
Nov 22 at 14:20
1
You’re missing that it’s already highly unlikely that you’ve flipped nine heads in a row.
– Randall
Nov 22 at 14:28
1
Let's go frequentist here. You sit down and flip ten coins and record the pattern of H and T. You are very unlikely to get HHHHHHHHHH don't you think? Do the experiment again, and again and again. On average, you will only get HHHHHHHHHH every 1024 attempts. Now think about your friend flipping the same coin 9 times but you can't see what he got. You throw the coin once to make it 10 times - how likely is a head on your coin?
– Paul
Nov 22 at 14:34
1
Following Paul's second comment, suppose you've done the 10-flip experiment a huge number of times, say a billion. Out of those billion runs, only about 1 in 1024 will be all heads. About 1 in 512 will begin with 9 heads. Of those that begin with 9 heads, about half will continue with a tenth head. That's what conditional probability 1/2 means.
– Andreas Blass
Nov 22 at 18:49
|
show 1 more comment
The chance of flipping 10 heads in a row is very unlikely don't you think? The chance of flipping any string of 10 heads or tails is $frac{1}{2^{10}}$ if you specify that string in advance.That is a completely different question from the chance of getting a heads on one throw (regardless of what you flipped before or the weather or anything else). There is no contradiction here.
– Paul
Nov 22 at 14:01
So given that I flipped 9 heads in a row, the chance that I flip 10 heads in a row is 1/2? And the chance of flipping 10 heads in a row is 1/1024?
– Kohler Fryer
Nov 22 at 14:20
1
You’re missing that it’s already highly unlikely that you’ve flipped nine heads in a row.
– Randall
Nov 22 at 14:28
1
Let's go frequentist here. You sit down and flip ten coins and record the pattern of H and T. You are very unlikely to get HHHHHHHHHH don't you think? Do the experiment again, and again and again. On average, you will only get HHHHHHHHHH every 1024 attempts. Now think about your friend flipping the same coin 9 times but you can't see what he got. You throw the coin once to make it 10 times - how likely is a head on your coin?
– Paul
Nov 22 at 14:34
1
Following Paul's second comment, suppose you've done the 10-flip experiment a huge number of times, say a billion. Out of those billion runs, only about 1 in 1024 will be all heads. About 1 in 512 will begin with 9 heads. Of those that begin with 9 heads, about half will continue with a tenth head. That's what conditional probability 1/2 means.
– Andreas Blass
Nov 22 at 18:49
The chance of flipping 10 heads in a row is very unlikely don't you think? The chance of flipping any string of 10 heads or tails is $frac{1}{2^{10}}$ if you specify that string in advance.That is a completely different question from the chance of getting a heads on one throw (regardless of what you flipped before or the weather or anything else). There is no contradiction here.
– Paul
Nov 22 at 14:01
The chance of flipping 10 heads in a row is very unlikely don't you think? The chance of flipping any string of 10 heads or tails is $frac{1}{2^{10}}$ if you specify that string in advance.That is a completely different question from the chance of getting a heads on one throw (regardless of what you flipped before or the weather or anything else). There is no contradiction here.
– Paul
Nov 22 at 14:01
So given that I flipped 9 heads in a row, the chance that I flip 10 heads in a row is 1/2? And the chance of flipping 10 heads in a row is 1/1024?
– Kohler Fryer
Nov 22 at 14:20
So given that I flipped 9 heads in a row, the chance that I flip 10 heads in a row is 1/2? And the chance of flipping 10 heads in a row is 1/1024?
– Kohler Fryer
Nov 22 at 14:20
1
1
You’re missing that it’s already highly unlikely that you’ve flipped nine heads in a row.
– Randall
Nov 22 at 14:28
You’re missing that it’s already highly unlikely that you’ve flipped nine heads in a row.
– Randall
Nov 22 at 14:28
1
1
Let's go frequentist here. You sit down and flip ten coins and record the pattern of H and T. You are very unlikely to get HHHHHHHHHH don't you think? Do the experiment again, and again and again. On average, you will only get HHHHHHHHHH every 1024 attempts. Now think about your friend flipping the same coin 9 times but you can't see what he got. You throw the coin once to make it 10 times - how likely is a head on your coin?
– Paul
Nov 22 at 14:34
Let's go frequentist here. You sit down and flip ten coins and record the pattern of H and T. You are very unlikely to get HHHHHHHHHH don't you think? Do the experiment again, and again and again. On average, you will only get HHHHHHHHHH every 1024 attempts. Now think about your friend flipping the same coin 9 times but you can't see what he got. You throw the coin once to make it 10 times - how likely is a head on your coin?
– Paul
Nov 22 at 14:34
1
1
Following Paul's second comment, suppose you've done the 10-flip experiment a huge number of times, say a billion. Out of those billion runs, only about 1 in 1024 will be all heads. About 1 in 512 will begin with 9 heads. Of those that begin with 9 heads, about half will continue with a tenth head. That's what conditional probability 1/2 means.
– Andreas Blass
Nov 22 at 18:49
Following Paul's second comment, suppose you've done the 10-flip experiment a huge number of times, say a billion. Out of those billion runs, only about 1 in 1024 will be all heads. About 1 in 512 will begin with 9 heads. Of those that begin with 9 heads, about half will continue with a tenth head. That's what conditional probability 1/2 means.
– Andreas Blass
Nov 22 at 18:49
|
show 1 more comment
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The chance of flipping 10 heads in a row is very unlikely don't you think? The chance of flipping any string of 10 heads or tails is $frac{1}{2^{10}}$ if you specify that string in advance.That is a completely different question from the chance of getting a heads on one throw (regardless of what you flipped before or the weather or anything else). There is no contradiction here.
– Paul
Nov 22 at 14:01
So given that I flipped 9 heads in a row, the chance that I flip 10 heads in a row is 1/2? And the chance of flipping 10 heads in a row is 1/1024?
– Kohler Fryer
Nov 22 at 14:20
1
You’re missing that it’s already highly unlikely that you’ve flipped nine heads in a row.
– Randall
Nov 22 at 14:28
1
Let's go frequentist here. You sit down and flip ten coins and record the pattern of H and T. You are very unlikely to get HHHHHHHHHH don't you think? Do the experiment again, and again and again. On average, you will only get HHHHHHHHHH every 1024 attempts. Now think about your friend flipping the same coin 9 times but you can't see what he got. You throw the coin once to make it 10 times - how likely is a head on your coin?
– Paul
Nov 22 at 14:34
1
Following Paul's second comment, suppose you've done the 10-flip experiment a huge number of times, say a billion. Out of those billion runs, only about 1 in 1024 will be all heads. About 1 in 512 will begin with 9 heads. Of those that begin with 9 heads, about half will continue with a tenth head. That's what conditional probability 1/2 means.
– Andreas Blass
Nov 22 at 18:49