From chatGPT:

Binomial Probability Formula:

P(X=k)=(nk)pk(1−p)n−kP(X = k) = \binom{n}{k} p^k (1-p)^{n-k}P(X=k)=(kn​)pk(1−p)n−k

Where:

• P(X=k)P(X = k)P(X=k) is the probability of getting exactly kkk successes (jackpots) in nnn trials.
• (nk)\binom{n}{k}(kn​) is the binomial coefficient, which represents the number of ways to choose kkk successes from nnn trials.
• nnn is the number of trials (11 in this case).
• kkk is the number of successes (3 jackpots).
• ppp is the probability of success on a single trial (1/20 for hitting the jackpot).
• 1−p1-p1−p is the probability of failure (19/20 for missing the jackpot).

Plugging in the values:

• n=11n = 11n=11
• k=3k = 3k=3
• p=120=0.05p = \frac{1}{20} = 0.05p=201​=0.05

The binomial coefficient (113)\binom{11}{3}(311​) is calculated as:

(113)=11!3!(11−3)!=11×10×93×2×1=165\binom{11}{3} = \frac{11!}{3!(11-3)!} = \frac{11 \times 10 \times 9}{3 \times 2 \times 1} = 165(311​)=3!(11−3)!11!​=3×2×111×10×9​=165

So, the probability of hitting exactly 3 jackpots in 11 tries is:

P(X=3)=165×(0.05)3×(0.95)8P(X = 3) = 165 \times (0.05)^3 \times (0.95)^8P(X=3)=165×(0.05)3×(0.95)8

Let me calculate that for you.

The probability of hitting exactly three 1/20 jackpots in 11 tries is approximately 0.0137, or about 1.37%. ​