Consider tossing a coin five times. What is the probability of getting the same $result$? Consider tossing a coin five times. What is the probability of getting the same $result$, on the $first \text{ two }$ or $last \text{ two }$ tosses?

The size of the sample space is $2^5.$
Let $E$ - first two tosses are same.
Let $F$ - last two tosses are same.

$P(E) + P(F)$
$|E| = |F| = 2^5$
$\text{_ _ _ _ _}$ Given 5 tosses, toss 1 can be heads or tails $\binom{2}{1}$. Toss 2 must be whatever toss 1 was. (1). Toss 3,4,5 can be any $\binom{2}{1}$$\binom{2}{1}$$\binom{2}{1}$.
So we have:
$$\frac{\binom{2}{1}(1)\binom{2}{1}\binom{2}{1}\binom{2}{1}}{2^5} = \frac{16}{32}$$
Similarly, for last two tosses same, it is also $$\frac{16}{32}$$

But this equals $1$, and $100\%$ is definitely not right ... what did I do wrong?

Clearly $E$ and $F$ are independent and each has probability $0.5$.
So the answer is $0.5+0.5-0.5^2=0.75$
