Multiple multiple choice (100 follower special)

There are $L$ persons who know the right answer, and $K$ persons who guess blindly. Of the second group, there are $k$ people who happened to guess correctly. The probability of guessing correctly from three answer options is $p = 1/3$ . The variable $k$ is binomial-distributed, so that the probability $P (k)$ for exactly $k$ correct answers is given by $P(k) = \frac{K!}{k! (K - k)!} p^{k} (1 - p)^{K - k}$ The expected value $\langle k \rangle$ and standard deviation $\sigma_k$ of this distribution are given by $\begin{aligned} \langle k \rangle &= K p \\ \sigma_k &= \sqrt{K p (1 - p)} \end{aligned}$ Suppose it is $L = K = 50$ . Then the numbers results to $\begin{aligned} \langle k \rangle &\approx 16.7 \\ \sigma_k &\approx 3.3 \end{aligned}$ In fact, there are $L + k = 63$ correct answers, so that in this case we have $k = 13$ . Thus, this value lies within the error interval $\langle k \rangle - \sigma_k <k <\langle k \rangle + \sigma_k$ and fits perfectly with our assumption $L = K = 50$ . Therefore, we expect that the numbers $L$ and $K$ are approximately equal. At least no statement can be made as to which of the two groups is larger.

Without any math, although statistically unlikely, we know that the 63 people could have all known the answer or could have all guessed, or anywhere in between. Therefore, we can't determine whether the majority knew.