Find the winning probability

The probability of one extraction is the same as that of one extraction. The essence of this problem is the application of lottery model.

Specifically, it is equivalent to 30 balls in a row, as long as the first five positions are studied: the first five positions are red balls, representing the first prize; There are four red balls and one black ball in the first five positions, representing the second prize; And there are three red balls and two black balls in the first five positions, which is the third prize.

Therefore, the probability of the first prize is: