Học sinh làm đúng k câu \(\left(0\le k\le50\right)\) và làm sai \(50-k\) câu
Xác suất làm đúng k câu là:
\(P\left(k\right)=\left(\dfrac{1}{4}\right)^k.\left(\dfrac{3}{4}\right)^{50-k}.C_{50}^k\)
\(\left\{{}\begin{matrix}P\left(k\right)\ge P\left(k-1\right)\\P\left(k\right)\ge P\left(k+1\right)\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}\left(\dfrac{1}{4}\right)^k\left(\dfrac{3}{4}\right)^{50-k}.\dfrac{50!}{\left(50-k\right)!k!}\ge\left(\dfrac{1}{4}\right)^{k-1}\left(\dfrac{3}{4}\right)^{51-k}.\dfrac{50!}{\left(51-k\right)!\left(k-1\right)!}\\\left(\dfrac{1}{4}\right)^k\left(\dfrac{3}{4}\right)^{50-k}.\dfrac{50!}{\left(50-k\right)!k!}\ge\left(\dfrac{1}{4}\right)^{k+1}\left(\dfrac{3}{4}\right)^{49-k}.\dfrac{50!}{\left(49-k\right)!\left(k+1\right)!}\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}4k\le51\\4k\ge47\end{matrix}\right.\) \(\Rightarrow k=12\)
