The lottery numbers drawn on February 28, 1969, were: 2, 34, 36, 50, 65. In connection with this, let's calculate the probability that in a lottery draw, the sum of the first two numbers, when arranged in ascending order, will be equal to the third number. What is the most frequent value of the third number in such draws?
All possible lottery draws are 5-element (without repetition) combinations of the numbers $1,2, \ldots, 90$, their number being

$$
N=\binom{90}{5}=43949268
$$

We calculate the number of draws that meet the given conditions, where the first three of the lottery numbers, arranged in increasing order, are denoted by $e, m, h$. We calculate the number of draws with the prescribed properties, where $h$ is a given value. Since $eK_{h}^{\prime}$ if $h30$, among the $K_{h}^{\prime}$ numbers, $K_{30}^{\prime}$ and $K_{31}^{\prime}$ are the largest, and $K_{30}^{\prime}=K_{31}^{\prime}$. We have seen that if $h$ is odd, $K_{h}=K_{h}^{\prime}$, and if $h$ is even, $K_{h}<K_{h}^{\prime}$, so among the $K_{h}$ numbers, $K_{31}$ is the largest. Therefore, among the draws we are examining, 31 is the most frequent value of the third number.

Martoni Viktor

Remarks. 1. In the 672 draws of the Hungarian lottery up to the end of 1969, the examined equality appeared 16 times. The ratio $16 / 672=1 / 42 \approx 0.024$ is in quite good agreement with the result of our calculation.

However, the 16 cases are too few to show any regularity in the frequency of the third number. The third numbers that occurred, arranged in increasing order, are: $7,12,13,15,15,15$ (in two cases as $7+8$), 26, $32,33,36,41,45,64,65,67,73$. The median used in the statistics (where as many data are larger as are smaller) is between 32 and 33.

2. We examined the case where the difference $k_{3}=h-(e+m)$ is 0. Among the 672 draws, $k_{3}$ was positive in 354 cases, negative in 302 cases, the largest value that occurred was 60, and the smallest -62. The number of occurrences of some values of $k_{3}$ around 0 is:

$$
\begin{array}{rrrrrrrrrr}
k_{3}=+1, & +2, & +3, & +4, & +5, & +6, & +7, & +8, & +9, & +10 \\
14, & 22, & 18, & 13, & 16, & 18, & 11, & 18, & 13, & 13 \\
k_{3}=-1, & -2, & -3, & -4, & -5, & -6, & -7, & -8, & -9, & -10 \\
16, & 18, & 16, & 18, & 17, & 10, & 11, & 11, & 10, & 15
\end{array}
$$