Probability that THHT occurs in a sequence of 10 coin tosses
$begingroup$
Assume we have a fair coin, and we throw the coin $10$ times in a row.
I want to calculate the probability that the sequence 'tail, head, head, tail' occurs.
So I think I can interpret this event as a binary number with $10$ digits. So $1$ means tail, $0$ means head. Therefore we have $2^{10} = 1024$ different outcomes of the $10$ throws. The sequence 'tail, head, head, tail' can start at $7$ different positions and so there are $7cdot2^6 = 448$ different outcomes of the $10$ throws with the sequence 'tail, head, head, tail'. So the probability would be $frac{448}{1024} = 0.4375$.
But I have a feeling there's something wrong?
probability combinatorics
New contributor
$endgroup$
add a comment |
$begingroup$
Assume we have a fair coin, and we throw the coin $10$ times in a row.
I want to calculate the probability that the sequence 'tail, head, head, tail' occurs.
So I think I can interpret this event as a binary number with $10$ digits. So $1$ means tail, $0$ means head. Therefore we have $2^{10} = 1024$ different outcomes of the $10$ throws. The sequence 'tail, head, head, tail' can start at $7$ different positions and so there are $7cdot2^6 = 448$ different outcomes of the $10$ throws with the sequence 'tail, head, head, tail'. So the probability would be $frac{448}{1024} = 0.4375$.
But I have a feeling there's something wrong?
probability combinatorics
New contributor
$endgroup$
2
$begingroup$
You are over-counting sequences like T H H T H H T .
$endgroup$
– kimchi lover
13 hours ago
1
$begingroup$
@kimchilover ...and even THHTHHTHHT.
$endgroup$
– CiaPan
12 hours ago
add a comment |
$begingroup$
Assume we have a fair coin, and we throw the coin $10$ times in a row.
I want to calculate the probability that the sequence 'tail, head, head, tail' occurs.
So I think I can interpret this event as a binary number with $10$ digits. So $1$ means tail, $0$ means head. Therefore we have $2^{10} = 1024$ different outcomes of the $10$ throws. The sequence 'tail, head, head, tail' can start at $7$ different positions and so there are $7cdot2^6 = 448$ different outcomes of the $10$ throws with the sequence 'tail, head, head, tail'. So the probability would be $frac{448}{1024} = 0.4375$.
But I have a feeling there's something wrong?
probability combinatorics
New contributor
$endgroup$
Assume we have a fair coin, and we throw the coin $10$ times in a row.
I want to calculate the probability that the sequence 'tail, head, head, tail' occurs.
So I think I can interpret this event as a binary number with $10$ digits. So $1$ means tail, $0$ means head. Therefore we have $2^{10} = 1024$ different outcomes of the $10$ throws. The sequence 'tail, head, head, tail' can start at $7$ different positions and so there are $7cdot2^6 = 448$ different outcomes of the $10$ throws with the sequence 'tail, head, head, tail'. So the probability would be $frac{448}{1024} = 0.4375$.
But I have a feeling there's something wrong?
probability combinatorics
probability combinatorics
New contributor
New contributor
edited 8 hours ago
Robert Howard
2,2383935
2,2383935
New contributor
asked 13 hours ago
geirageira
663
663
New contributor
New contributor
2
$begingroup$
You are over-counting sequences like T H H T H H T .
$endgroup$
– kimchi lover
13 hours ago
1
$begingroup$
@kimchilover ...and even THHTHHTHHT.
$endgroup$
– CiaPan
12 hours ago
add a comment |
2
$begingroup$
You are over-counting sequences like T H H T H H T .
$endgroup$
– kimchi lover
13 hours ago
1
$begingroup$
@kimchilover ...and even THHTHHTHHT.
$endgroup$
– CiaPan
12 hours ago
2
2
$begingroup$
You are over-counting sequences like T H H T H H T .
$endgroup$
– kimchi lover
13 hours ago
$begingroup$
You are over-counting sequences like T H H T H H T .
$endgroup$
– kimchi lover
13 hours ago
1
1
$begingroup$
@kimchilover ...and even THHTHHTHHT.
$endgroup$
– CiaPan
12 hours ago
$begingroup$
@kimchilover ...and even THHTHHTHHT.
$endgroup$
– CiaPan
12 hours ago
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
You can use the Inclusion-exclusion principle to solve this. When I do this I get $$7cdot 2^6 - 6cdot 2^2 - 4cdot 2^3+1$$ where the first term is what you get, where the second and third terms count the number of sequences with two non-overlapping instances of T H H T and of sequences with one overlap, like T H H T H H T, and finally the number of sequences with a triple overlap, T H H T H H T H H T.
Confession: I has earlier got $45cdot2^2$ for the second term, by a mental blunder, as AnnaSaabel pointed out. There are 2 "gaps" to separate the two instances of THHT, which can occur before, between, or after the 2 instances; they can be distributed in any of the 6 ways 200,020,002,110,101, or 011.
Added: if the number of coin tosses were $n=100$ (say), and the pattern sought was still THHT, this method becomes clumsy. A different method is to construct a Markov chain with states representing how far a string matching algorithm has progressed in matching the given pattern. If $M$ is the transition matrix for this chain, the desired answer is the entry in the matrix $M^n$ corresponing to the pair $(text{start state}, text{accepting state})$.
$endgroup$
1
$begingroup$
How did you calculate 45 for the second term?
$endgroup$
– Anna Saabel
11 hours ago
$begingroup$
@AnnaSaabel By a mental blunder. Thanks for the catch!
$endgroup$
– kimchi lover
11 hours ago
1
$begingroup$
This is correct, as this Python 3 code snippet gives the same number: $tag*{}$print(sum('0110' in s for s in [bin(n)[-10:] for n in range(1<<10,1<<11)]))
$endgroup$
– Mike Earnest
10 hours ago
$begingroup$
Your transition matrix is wrong: a match failure in state 1 or 2 will result in state 1, not state 0. In general, the transition matrix is basically the same thing as the array produced in the preprocessing step of KMP algorithm.
$endgroup$
– infmagic2047
10 hours ago
$begingroup$
@infmagic2047 You are right; I've hopefully corrected this.
$endgroup$
– kimchi lover
10 hours ago
add a comment |
$begingroup$
As @kimchilover states in the comments, you are counting some 10-digit binary numbers more than once in the number $7cdot 2^6$. To make this more obvious, consider a different problem: to find the probability that the sequence 'heads' appears. By your counting logic, there are 10 places for it to begin, so there are $10cdot 2^9$ different outcomes of the 10 throws with the sequence 'heads', so the probability would be $frac{10cdot 2^9}{2^{10}} = 5$. That can't be good. It's very clear now that the issue is overcounting -- you have five times as many sequences with 'heads' in them as the number of sequences total! The problem is that we have counted sequences with multiple heads many times. For example, the sequence of all heads is counted $10$ times, once for each of the places where the sequence 'heads' begins within it.
As I write this, I see that @kimchilover also just posted an answer to the question which directs you to the inclusion-exclusion principle, so I'll stop here with an answer which could just help you to try generalizing arguments which feel fishy to see where they go wrong. Good job detecting the fishiness!
$endgroup$
add a comment |
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2 Answers
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$begingroup$
You can use the Inclusion-exclusion principle to solve this. When I do this I get $$7cdot 2^6 - 6cdot 2^2 - 4cdot 2^3+1$$ where the first term is what you get, where the second and third terms count the number of sequences with two non-overlapping instances of T H H T and of sequences with one overlap, like T H H T H H T, and finally the number of sequences with a triple overlap, T H H T H H T H H T.
Confession: I has earlier got $45cdot2^2$ for the second term, by a mental blunder, as AnnaSaabel pointed out. There are 2 "gaps" to separate the two instances of THHT, which can occur before, between, or after the 2 instances; they can be distributed in any of the 6 ways 200,020,002,110,101, or 011.
Added: if the number of coin tosses were $n=100$ (say), and the pattern sought was still THHT, this method becomes clumsy. A different method is to construct a Markov chain with states representing how far a string matching algorithm has progressed in matching the given pattern. If $M$ is the transition matrix for this chain, the desired answer is the entry in the matrix $M^n$ corresponing to the pair $(text{start state}, text{accepting state})$.
$endgroup$
1
$begingroup$
How did you calculate 45 for the second term?
$endgroup$
– Anna Saabel
11 hours ago
$begingroup$
@AnnaSaabel By a mental blunder. Thanks for the catch!
$endgroup$
– kimchi lover
11 hours ago
1
$begingroup$
This is correct, as this Python 3 code snippet gives the same number: $tag*{}$print(sum('0110' in s for s in [bin(n)[-10:] for n in range(1<<10,1<<11)]))
$endgroup$
– Mike Earnest
10 hours ago
$begingroup$
Your transition matrix is wrong: a match failure in state 1 or 2 will result in state 1, not state 0. In general, the transition matrix is basically the same thing as the array produced in the preprocessing step of KMP algorithm.
$endgroup$
– infmagic2047
10 hours ago
$begingroup$
@infmagic2047 You are right; I've hopefully corrected this.
$endgroup$
– kimchi lover
10 hours ago
add a comment |
$begingroup$
You can use the Inclusion-exclusion principle to solve this. When I do this I get $$7cdot 2^6 - 6cdot 2^2 - 4cdot 2^3+1$$ where the first term is what you get, where the second and third terms count the number of sequences with two non-overlapping instances of T H H T and of sequences with one overlap, like T H H T H H T, and finally the number of sequences with a triple overlap, T H H T H H T H H T.
Confession: I has earlier got $45cdot2^2$ for the second term, by a mental blunder, as AnnaSaabel pointed out. There are 2 "gaps" to separate the two instances of THHT, which can occur before, between, or after the 2 instances; they can be distributed in any of the 6 ways 200,020,002,110,101, or 011.
Added: if the number of coin tosses were $n=100$ (say), and the pattern sought was still THHT, this method becomes clumsy. A different method is to construct a Markov chain with states representing how far a string matching algorithm has progressed in matching the given pattern. If $M$ is the transition matrix for this chain, the desired answer is the entry in the matrix $M^n$ corresponing to the pair $(text{start state}, text{accepting state})$.
$endgroup$
1
$begingroup$
How did you calculate 45 for the second term?
$endgroup$
– Anna Saabel
11 hours ago
$begingroup$
@AnnaSaabel By a mental blunder. Thanks for the catch!
$endgroup$
– kimchi lover
11 hours ago
1
$begingroup$
This is correct, as this Python 3 code snippet gives the same number: $tag*{}$print(sum('0110' in s for s in [bin(n)[-10:] for n in range(1<<10,1<<11)]))
$endgroup$
– Mike Earnest
10 hours ago
$begingroup$
Your transition matrix is wrong: a match failure in state 1 or 2 will result in state 1, not state 0. In general, the transition matrix is basically the same thing as the array produced in the preprocessing step of KMP algorithm.
$endgroup$
– infmagic2047
10 hours ago
$begingroup$
@infmagic2047 You are right; I've hopefully corrected this.
$endgroup$
– kimchi lover
10 hours ago
add a comment |
$begingroup$
You can use the Inclusion-exclusion principle to solve this. When I do this I get $$7cdot 2^6 - 6cdot 2^2 - 4cdot 2^3+1$$ where the first term is what you get, where the second and third terms count the number of sequences with two non-overlapping instances of T H H T and of sequences with one overlap, like T H H T H H T, and finally the number of sequences with a triple overlap, T H H T H H T H H T.
Confession: I has earlier got $45cdot2^2$ for the second term, by a mental blunder, as AnnaSaabel pointed out. There are 2 "gaps" to separate the two instances of THHT, which can occur before, between, or after the 2 instances; they can be distributed in any of the 6 ways 200,020,002,110,101, or 011.
Added: if the number of coin tosses were $n=100$ (say), and the pattern sought was still THHT, this method becomes clumsy. A different method is to construct a Markov chain with states representing how far a string matching algorithm has progressed in matching the given pattern. If $M$ is the transition matrix for this chain, the desired answer is the entry in the matrix $M^n$ corresponing to the pair $(text{start state}, text{accepting state})$.
$endgroup$
You can use the Inclusion-exclusion principle to solve this. When I do this I get $$7cdot 2^6 - 6cdot 2^2 - 4cdot 2^3+1$$ where the first term is what you get, where the second and third terms count the number of sequences with two non-overlapping instances of T H H T and of sequences with one overlap, like T H H T H H T, and finally the number of sequences with a triple overlap, T H H T H H T H H T.
Confession: I has earlier got $45cdot2^2$ for the second term, by a mental blunder, as AnnaSaabel pointed out. There are 2 "gaps" to separate the two instances of THHT, which can occur before, between, or after the 2 instances; they can be distributed in any of the 6 ways 200,020,002,110,101, or 011.
Added: if the number of coin tosses were $n=100$ (say), and the pattern sought was still THHT, this method becomes clumsy. A different method is to construct a Markov chain with states representing how far a string matching algorithm has progressed in matching the given pattern. If $M$ is the transition matrix for this chain, the desired answer is the entry in the matrix $M^n$ corresponing to the pair $(text{start state}, text{accepting state})$.
edited 10 hours ago
answered 12 hours ago
kimchi loverkimchi lover
11.4k31229
11.4k31229
1
$begingroup$
How did you calculate 45 for the second term?
$endgroup$
– Anna Saabel
11 hours ago
$begingroup$
@AnnaSaabel By a mental blunder. Thanks for the catch!
$endgroup$
– kimchi lover
11 hours ago
1
$begingroup$
This is correct, as this Python 3 code snippet gives the same number: $tag*{}$print(sum('0110' in s for s in [bin(n)[-10:] for n in range(1<<10,1<<11)]))
$endgroup$
– Mike Earnest
10 hours ago
$begingroup$
Your transition matrix is wrong: a match failure in state 1 or 2 will result in state 1, not state 0. In general, the transition matrix is basically the same thing as the array produced in the preprocessing step of KMP algorithm.
$endgroup$
– infmagic2047
10 hours ago
$begingroup$
@infmagic2047 You are right; I've hopefully corrected this.
$endgroup$
– kimchi lover
10 hours ago
add a comment |
1
$begingroup$
How did you calculate 45 for the second term?
$endgroup$
– Anna Saabel
11 hours ago
$begingroup$
@AnnaSaabel By a mental blunder. Thanks for the catch!
$endgroup$
– kimchi lover
11 hours ago
1
$begingroup$
This is correct, as this Python 3 code snippet gives the same number: $tag*{}$print(sum('0110' in s for s in [bin(n)[-10:] for n in range(1<<10,1<<11)]))
$endgroup$
– Mike Earnest
10 hours ago
$begingroup$
Your transition matrix is wrong: a match failure in state 1 or 2 will result in state 1, not state 0. In general, the transition matrix is basically the same thing as the array produced in the preprocessing step of KMP algorithm.
$endgroup$
– infmagic2047
10 hours ago
$begingroup$
@infmagic2047 You are right; I've hopefully corrected this.
$endgroup$
– kimchi lover
10 hours ago
1
1
$begingroup$
How did you calculate 45 for the second term?
$endgroup$
– Anna Saabel
11 hours ago
$begingroup$
How did you calculate 45 for the second term?
$endgroup$
– Anna Saabel
11 hours ago
$begingroup$
@AnnaSaabel By a mental blunder. Thanks for the catch!
$endgroup$
– kimchi lover
11 hours ago
$begingroup$
@AnnaSaabel By a mental blunder. Thanks for the catch!
$endgroup$
– kimchi lover
11 hours ago
1
1
$begingroup$
This is correct, as this Python 3 code snippet gives the same number: $tag*{}$
print(sum('0110' in s for s in [bin(n)[-10:] for n in range(1<<10,1<<11)]))
$endgroup$
– Mike Earnest
10 hours ago
$begingroup$
This is correct, as this Python 3 code snippet gives the same number: $tag*{}$
print(sum('0110' in s for s in [bin(n)[-10:] for n in range(1<<10,1<<11)]))
$endgroup$
– Mike Earnest
10 hours ago
$begingroup$
Your transition matrix is wrong: a match failure in state 1 or 2 will result in state 1, not state 0. In general, the transition matrix is basically the same thing as the array produced in the preprocessing step of KMP algorithm.
$endgroup$
– infmagic2047
10 hours ago
$begingroup$
Your transition matrix is wrong: a match failure in state 1 or 2 will result in state 1, not state 0. In general, the transition matrix is basically the same thing as the array produced in the preprocessing step of KMP algorithm.
$endgroup$
– infmagic2047
10 hours ago
$begingroup$
@infmagic2047 You are right; I've hopefully corrected this.
$endgroup$
– kimchi lover
10 hours ago
$begingroup$
@infmagic2047 You are right; I've hopefully corrected this.
$endgroup$
– kimchi lover
10 hours ago
add a comment |
$begingroup$
As @kimchilover states in the comments, you are counting some 10-digit binary numbers more than once in the number $7cdot 2^6$. To make this more obvious, consider a different problem: to find the probability that the sequence 'heads' appears. By your counting logic, there are 10 places for it to begin, so there are $10cdot 2^9$ different outcomes of the 10 throws with the sequence 'heads', so the probability would be $frac{10cdot 2^9}{2^{10}} = 5$. That can't be good. It's very clear now that the issue is overcounting -- you have five times as many sequences with 'heads' in them as the number of sequences total! The problem is that we have counted sequences with multiple heads many times. For example, the sequence of all heads is counted $10$ times, once for each of the places where the sequence 'heads' begins within it.
As I write this, I see that @kimchilover also just posted an answer to the question which directs you to the inclusion-exclusion principle, so I'll stop here with an answer which could just help you to try generalizing arguments which feel fishy to see where they go wrong. Good job detecting the fishiness!
$endgroup$
add a comment |
$begingroup$
As @kimchilover states in the comments, you are counting some 10-digit binary numbers more than once in the number $7cdot 2^6$. To make this more obvious, consider a different problem: to find the probability that the sequence 'heads' appears. By your counting logic, there are 10 places for it to begin, so there are $10cdot 2^9$ different outcomes of the 10 throws with the sequence 'heads', so the probability would be $frac{10cdot 2^9}{2^{10}} = 5$. That can't be good. It's very clear now that the issue is overcounting -- you have five times as many sequences with 'heads' in them as the number of sequences total! The problem is that we have counted sequences with multiple heads many times. For example, the sequence of all heads is counted $10$ times, once for each of the places where the sequence 'heads' begins within it.
As I write this, I see that @kimchilover also just posted an answer to the question which directs you to the inclusion-exclusion principle, so I'll stop here with an answer which could just help you to try generalizing arguments which feel fishy to see where they go wrong. Good job detecting the fishiness!
$endgroup$
add a comment |
$begingroup$
As @kimchilover states in the comments, you are counting some 10-digit binary numbers more than once in the number $7cdot 2^6$. To make this more obvious, consider a different problem: to find the probability that the sequence 'heads' appears. By your counting logic, there are 10 places for it to begin, so there are $10cdot 2^9$ different outcomes of the 10 throws with the sequence 'heads', so the probability would be $frac{10cdot 2^9}{2^{10}} = 5$. That can't be good. It's very clear now that the issue is overcounting -- you have five times as many sequences with 'heads' in them as the number of sequences total! The problem is that we have counted sequences with multiple heads many times. For example, the sequence of all heads is counted $10$ times, once for each of the places where the sequence 'heads' begins within it.
As I write this, I see that @kimchilover also just posted an answer to the question which directs you to the inclusion-exclusion principle, so I'll stop here with an answer which could just help you to try generalizing arguments which feel fishy to see where they go wrong. Good job detecting the fishiness!
$endgroup$
As @kimchilover states in the comments, you are counting some 10-digit binary numbers more than once in the number $7cdot 2^6$. To make this more obvious, consider a different problem: to find the probability that the sequence 'heads' appears. By your counting logic, there are 10 places for it to begin, so there are $10cdot 2^9$ different outcomes of the 10 throws with the sequence 'heads', so the probability would be $frac{10cdot 2^9}{2^{10}} = 5$. That can't be good. It's very clear now that the issue is overcounting -- you have five times as many sequences with 'heads' in them as the number of sequences total! The problem is that we have counted sequences with multiple heads many times. For example, the sequence of all heads is counted $10$ times, once for each of the places where the sequence 'heads' begins within it.
As I write this, I see that @kimchilover also just posted an answer to the question which directs you to the inclusion-exclusion principle, so I'll stop here with an answer which could just help you to try generalizing arguments which feel fishy to see where they go wrong. Good job detecting the fishiness!
edited 12 hours ago
answered 12 hours ago
cspruncsprun
2,325210
2,325210
add a comment |
add a comment |
geira is a new contributor. Be nice, and check out our Code of Conduct.
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2
$begingroup$
You are over-counting sequences like T H H T H H T .
$endgroup$
– kimchi lover
13 hours ago
1
$begingroup$
@kimchilover ...and even THHTHHTHHT.
$endgroup$
– CiaPan
12 hours ago