14. Pills (from 9th grade. 2 points). The Absent-Minded Scientist had a sore knee. The doctor prescribed him 10 pills for his knee: to take one pill daily. The pills help in $90 \%$ of cases, and in $2 \%$ of cases, there is a side effect - absent-mindedness disappears if it was present.

Another doctor prescribed the Scientist pills for absent-mindedness - also one per day for 10 days in a row. These pills cure absent-mindedness in $80 \%$ of cases, but in $5 \%$ of cases, there is a side effect - the knees start to hurt.

The bottles of pills look similar, and when the Scientist went on a ten-day business trip, he took one bottle with him but completely ignored which one. For ten days, he took one pill a day and returned completely healthy. Both the absent-mindedness and the sore knee were gone. Find the probability that the Scientist took the bottle of pills for absent-mindedness.

![](https://cdn.mathpix.com/cropped/2024_05_06_677ee29ca0a1bca903d6g-10.jpg?height=445&width=645&top_left_y=1037&top_left_x=1274)
Solution. Let's consider the events $R$ "The scientist took the pills for absent-mindedness", $A$ "The knee stopped hurting" and $B$ "Absent-mindedness disappeared".

We need to find the conditional probability $\mathrm{P}(R \mid A \cap B)$:

$$
\begin{gathered}
\mathrm{P}(R \mid A \cap B)=\frac{\mathrm{P}(R \cap A \cap B)}{\mathrm{P}(A \cap B)}=\frac{\mathrm{P}(R \cap A \cap B)}{\mathrm{P}(R \cap A \cap B)+\mathrm{P}(\bar{R} \cap A \cap B)}= \\
=\frac{\frac{1}{2} \cdot 0.8 \cdot 0.05}{\frac{1}{2} \cdot 0.8 \cdot 0.05+\frac{1}{2} \cdot 0.9 \cdot 0.02}=\frac{0.04}{0.04+0.018} \approx 0.69
\end{gathered}
$$

Answer: approximately 0.69.