Not sure if this was answered already but this is how the math checks out:
The probability they're trying to define here that's not well translated is "the probability of at least 1 success in gacha." So this might mean 1 win, or 2 prize wins or 3 prize wins, up to 20 prize wins since he's doing gacha total of 20 times. The number of attempts can be defined as 20 attempts (n=20). The easiest way to calculate the probability of "1 win chance"+"2 win chance"+"3 win chance"...+"20 win chance" is by taking the total probability of everything (the two outcomes are win and lose, simply put the formula 'all outcome'='win chance'+'lose chance'=100%=1) subtracted by the chance of losing. 100%-lose=win or 1-lose=win, where we define win as at least 1 success and losing as a total of 0 successes out of 20 tries.
To calculate the probability of losing, multiply each outcome by each other. Let's say you're trying to flip a coin and we consider tails to be loss and heads to be win. To calculate 3 losses, we do (0.5)*(0.5)*(0.5)=1/8. Another way to look at this is (1/2)^3 or ('chance of loss')^n where n=3 in this example since we flip the coin 3 times. Applying this to gacha, the chance of losing is (0.95)^20=0.358 or rounded to 36%.
Thus, the protagonist is stating that since he gets 0 wins 36% of the time, he will get at least 1 win (up to 20 wins) 64% of the time. He is pointing out that 5% * 20 is not the correct way to applying probability of outcomes in this situation. This is also why +5 atk or +15 atk work gloves were worth so fucking much in maplestory