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The Question

What is the probability that a binary string of length 16, chosen uniformly from all binary strings of length 16, has exactly eight 1s.

My Work

There are $2^{16}$ binary strings of length 16. Therefore, our sample space is $2^{16}$.

We can make a binary string of length 16 like so: Select 8 spaces to place our $1$s $\binom{16}{8}$ ways to do this, and then fill the remaining spaces with $0$s.

Therefore, the probability of our event = $\frac{\binom{16}{8}}{2^{16}}$

My Question

I don't have a solution for this question available (Checked book and internet). Wondering if what I did was correct

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  • 5
    $\begingroup$ It is just fine :). $\endgroup$ – Phicar Jan 18 '15 at 4:44
  • $\begingroup$ Yep, it's almost $20\%$ $\endgroup$ – orangeskid Jan 18 '15 at 6:53
  • $\begingroup$ Yes, $\frac{6435}{32768}$ $\endgroup$ – Joffan Jan 19 '15 at 18:16
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Community wiki answer so the question can be marked as answered:

As Phicar remarked and 5 upvoters confirmed, your calculation is correct.

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