15.6.16 Definition and properties of probability

Problem: There are \( S \) black and \( N-S \) white balls in the box. Let \( T_{1} \) be the extraction number, in case of which a black ball appears for the 1st time (i.e., \( T_{1} \) is the waiting time). Let's show that \( T_{1} \) has a geometric distribution (the ball, taken out, is returned) and calculate the expected value and the dispersion of \( T_{1} \).