37. A regular triangle is inscribed in a circle of radius $R$. What is the probability that four randomly placed points within this circle will be inside the triangle?
Solution. Consider the events:

$A_{i}$ - a randomly placed point in the circle will be inside the equilateral triangle inscribed in this circle ( $i=1,2,3,4$ ).

B - four randomly placed points in the circle will be inside the equilateral triangle inscribed in this circle.

Event $B$ consists of the first, second, third, and fourth points all being inside the triangle. This means that event $B$ is the product of events $A_{1}: B=A_{1} A_{2} A_{3} A_{4}$.

$A_{1}, A_{2}, A_{3}, A_{4}$ are independent events with equal probabilities. Let: $P\left(A_{i}\right)=p$. According to the solution of problem 20, the probability $p=\frac{3 \sqrt{3}}{4 \pi}$.

The probability of event $B$ will be found using the multiplication theorem of probabilities of independent events. Applying formula (15), we get

$$
P(B)=p^{4}=\left(\frac{3 \sqrt{3}}{4 \pi}\right)^{4} \approx 0.029
$$