If I randomly choose 2 bitstring on length (n) with n to be an even number, what is the probability, parametrized on n, that at least n/2 of the bits are equals?

In my mind, since the random choice of every single bit is considered independent, I presume that the total probability is 1/2, but i'm sincerely confused.

Edit: My English is bad, let me give a little example

If bistring_1(4) = 1001 and bitstring_2(4) = 1010 the statement is true because they have in common at least n/2 bits.

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  • 2
    $\begingroup$ If all strings are equally likely and independent and $n$ is even then the probability that two match exactly $n/2$ times is $\frac{n!}{((n/2)!)^2 2^n}$. The probability they match more than $n/2$ times is half of the remaining probability. Add these together and you get slightly more than $\frac12$ $\endgroup$
    – Henry
    Nov 25 at 15:10

If we assume that all bitstrings are equally likely then the problem is quite simple. Each matched pair of bits in the bitstrings will match with fixed probability $\tfrac{1}{2}$ and the number of matches has distribution:

$$M \sim \text{Bin}(n, \tfrac{1}{2}).$$

Consequently, the probability of at least $n/2$ matches (for even $n$) is:

$$\begin{align} \mathbb{P}(\text{At least } \tfrac{n}{2} \text{ matches}) &= \sum_{m=n/2}^n \text{Bin}(m|n, \tfrac{1}{2}) \\[6pt] &= \frac{1}{2^n} \sum_{m=n/2}^n {n \choose m} \\[6pt] &= \frac{1}{2} \Bigg[1 + \frac{1}{2^n} {n \choose n/2} \Bigg]. \\[6pt] \end{align}$$


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