My statistics knowledge is a little rusty, so I will be happy for any help.
For fun I've downloaded the app ESP trainer.
It shows you 4 colors, of which you must choose the correct one (which has been chosen randomly by the app). If you choose a wrong one, it will continue to the next question with a new correct color (so, only 1 guess per question).
It shows you 24 consecutive questions, and the correct answers are of course assumed to be independent of each other.
What is the probability of getting 11 answers correct out of 24 questions?
Or even better, what is the probability of getting x answers out of 24 questions correct?
Best Answer
If you set $X=$ the number of correct answers (when solving problems with the binomial distribution, the word choice is often "successes"), then $X$ follows the binomial distribution with parameter value $p=\frac{1}{4}$. The probabilities are relatively easy to calculate. For example, the probability that there are exactly $11$ successes is $$ P(X = 11) = {24 \choose 11} \left(\frac{1}{4} \right)^{11} \left(\frac{3}{4} \right)^{24-11} \approx 0.014~139 $$ Or about $1.4~\%$. On the other hand, the probability of at least 11 successes, is $$ \begin{split} P(X \geq 11) &= P(X = 11) + P(X = 12) + \ldots + P(X = 24)\\ &= \sum_{k=11}^{24} {24 \choose k} \left(\frac{1}{4} \right)^k \left(\frac{3}{4} \right)^{24-k}\\ &\approx 0.021~338 \end{split} $$
PS. I don't recommend thinking in terms of "1 out of 47 trials", because then we're mixing up two different notions of a "trial" in this context. So just stick to thinking "about two per cent".