# Solution Verification: Probability That a Binary String of length $16$ has exactly eight $1$s

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

• It is just fine :). – Phicar Jan 18 '15 at 4:44
• Yep, it's almost $20\%$ – Orest Bucicovschi Jan 18 '15 at 6:53
• Yes, $\frac{6435}{32768}$ – Joffan Jan 19 '15 at 18:16

## 1 Answer

Community wiki answer so the question can be marked as answered:

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