Probability and counting bit string
$begingroup$
How many 10-bit strings are there subject to each of the following restrictions if the first two bits are the same as the last two bits?
My answer:
1) So for the first $2$ bits and the last $2$ bits we only have $2$ choices: either those $4$ bits are all 0s or all 1s. So $2$ choices.
2) Then we have $(10-2-2)$ $6$ bits left with $2$ choices so $6^2$.
Conclusion : $2 * 2^6$ or $2^7$
But, I just read that the solution is $2^8$ I don't understand why since 4 bits should be exactly the same
thanks
probability discrete-mathematics
$endgroup$
add a comment |
$begingroup$
How many 10-bit strings are there subject to each of the following restrictions if the first two bits are the same as the last two bits?
My answer:
1) So for the first $2$ bits and the last $2$ bits we only have $2$ choices: either those $4$ bits are all 0s or all 1s. So $2$ choices.
2) Then we have $(10-2-2)$ $6$ bits left with $2$ choices so $6^2$.
Conclusion : $2 * 2^6$ or $2^7$
But, I just read that the solution is $2^8$ I don't understand why since 4 bits should be exactly the same
thanks
probability discrete-mathematics
$endgroup$
$begingroup$
In step 1, you have two choices for each of the first two bits. Choosing the first two bits also determines the last two bits since they are the same. In step 2, you should have written $2^6$ since you have two choices for each of the six middle bits.
$endgroup$
– N. F. Taussig
Nov 24 '18 at 23:36
add a comment |
$begingroup$
How many 10-bit strings are there subject to each of the following restrictions if the first two bits are the same as the last two bits?
My answer:
1) So for the first $2$ bits and the last $2$ bits we only have $2$ choices: either those $4$ bits are all 0s or all 1s. So $2$ choices.
2) Then we have $(10-2-2)$ $6$ bits left with $2$ choices so $6^2$.
Conclusion : $2 * 2^6$ or $2^7$
But, I just read that the solution is $2^8$ I don't understand why since 4 bits should be exactly the same
thanks
probability discrete-mathematics
$endgroup$
How many 10-bit strings are there subject to each of the following restrictions if the first two bits are the same as the last two bits?
My answer:
1) So for the first $2$ bits and the last $2$ bits we only have $2$ choices: either those $4$ bits are all 0s or all 1s. So $2$ choices.
2) Then we have $(10-2-2)$ $6$ bits left with $2$ choices so $6^2$.
Conclusion : $2 * 2^6$ or $2^7$
But, I just read that the solution is $2^8$ I don't understand why since 4 bits should be exactly the same
thanks
probability discrete-mathematics
probability discrete-mathematics
edited Nov 24 '18 at 23:50
Key Flex
7,77441232
7,77441232
asked Nov 24 '18 at 22:40
Laura1999Laura1999
202
202
$begingroup$
In step 1, you have two choices for each of the first two bits. Choosing the first two bits also determines the last two bits since they are the same. In step 2, you should have written $2^6$ since you have two choices for each of the six middle bits.
$endgroup$
– N. F. Taussig
Nov 24 '18 at 23:36
add a comment |
$begingroup$
In step 1, you have two choices for each of the first two bits. Choosing the first two bits also determines the last two bits since they are the same. In step 2, you should have written $2^6$ since you have two choices for each of the six middle bits.
$endgroup$
– N. F. Taussig
Nov 24 '18 at 23:36
$begingroup$
In step 1, you have two choices for each of the first two bits. Choosing the first two bits also determines the last two bits since they are the same. In step 2, you should have written $2^6$ since you have two choices for each of the six middle bits.
$endgroup$
– N. F. Taussig
Nov 24 '18 at 23:36
$begingroup$
In step 1, you have two choices for each of the first two bits. Choosing the first two bits also determines the last two bits since they are the same. In step 2, you should have written $2^6$ since you have two choices for each of the six middle bits.
$endgroup$
– N. F. Taussig
Nov 24 '18 at 23:36
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
I think you misinterpreted "first two bits are the same as the last two bits."
It is possible that the first two bits are 01, and that the last two bits are 01, for example. Thus there are $4$ possibilities for these four bits, and then multiplied by $2^6$ for the remaining six bits yields $2^8$.
$endgroup$
add a comment |
$begingroup$
If the last two bits have to be the same as the first two bits, then effectively you can only vary the first $8$ bits. So: $2^8$possible bit strings
$endgroup$
add a comment |
$begingroup$
Each of the first $8$ bits can be either $0$ or $1$. Once the the first $8$ bits are determined, the last two
bits must match the first two bits, so there are no remaining choices for the string. Thus, the number of strings in which the first two bits are the same as the last two bits is $2^8$
$endgroup$
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
I think you misinterpreted "first two bits are the same as the last two bits."
It is possible that the first two bits are 01, and that the last two bits are 01, for example. Thus there are $4$ possibilities for these four bits, and then multiplied by $2^6$ for the remaining six bits yields $2^8$.
$endgroup$
add a comment |
$begingroup$
I think you misinterpreted "first two bits are the same as the last two bits."
It is possible that the first two bits are 01, and that the last two bits are 01, for example. Thus there are $4$ possibilities for these four bits, and then multiplied by $2^6$ for the remaining six bits yields $2^8$.
$endgroup$
add a comment |
$begingroup$
I think you misinterpreted "first two bits are the same as the last two bits."
It is possible that the first two bits are 01, and that the last two bits are 01, for example. Thus there are $4$ possibilities for these four bits, and then multiplied by $2^6$ for the remaining six bits yields $2^8$.
$endgroup$
I think you misinterpreted "first two bits are the same as the last two bits."
It is possible that the first two bits are 01, and that the last two bits are 01, for example. Thus there are $4$ possibilities for these four bits, and then multiplied by $2^6$ for the remaining six bits yields $2^8$.
answered Nov 24 '18 at 22:50
angryavianangryavian
40k23280
40k23280
add a comment |
add a comment |
$begingroup$
If the last two bits have to be the same as the first two bits, then effectively you can only vary the first $8$ bits. So: $2^8$possible bit strings
$endgroup$
add a comment |
$begingroup$
If the last two bits have to be the same as the first two bits, then effectively you can only vary the first $8$ bits. So: $2^8$possible bit strings
$endgroup$
add a comment |
$begingroup$
If the last two bits have to be the same as the first two bits, then effectively you can only vary the first $8$ bits. So: $2^8$possible bit strings
$endgroup$
If the last two bits have to be the same as the first two bits, then effectively you can only vary the first $8$ bits. So: $2^8$possible bit strings
answered Nov 24 '18 at 23:07
Bram28Bram28
60.5k44590
60.5k44590
add a comment |
add a comment |
$begingroup$
Each of the first $8$ bits can be either $0$ or $1$. Once the the first $8$ bits are determined, the last two
bits must match the first two bits, so there are no remaining choices for the string. Thus, the number of strings in which the first two bits are the same as the last two bits is $2^8$
$endgroup$
add a comment |
$begingroup$
Each of the first $8$ bits can be either $0$ or $1$. Once the the first $8$ bits are determined, the last two
bits must match the first two bits, so there are no remaining choices for the string. Thus, the number of strings in which the first two bits are the same as the last two bits is $2^8$
$endgroup$
add a comment |
$begingroup$
Each of the first $8$ bits can be either $0$ or $1$. Once the the first $8$ bits are determined, the last two
bits must match the first two bits, so there are no remaining choices for the string. Thus, the number of strings in which the first two bits are the same as the last two bits is $2^8$
$endgroup$
Each of the first $8$ bits can be either $0$ or $1$. Once the the first $8$ bits are determined, the last two
bits must match the first two bits, so there are no remaining choices for the string. Thus, the number of strings in which the first two bits are the same as the last two bits is $2^8$
answered Nov 24 '18 at 23:49
Key FlexKey Flex
7,77441232
7,77441232
add a comment |
add a comment |
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$begingroup$
In step 1, you have two choices for each of the first two bits. Choosing the first two bits also determines the last two bits since they are the same. In step 2, you should have written $2^6$ since you have two choices for each of the six middle bits.
$endgroup$
– N. F. Taussig
Nov 24 '18 at 23:36