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You are going to play penny slots at a casino. The slot machine you have picked has 4 reels with 20 different symbols on each reel.

a) What is the probability of the same symbol appearing on all of the reels?

b) What is the probability of winning 1 dollar in the slot machine after spending 5 dollars?

For a), I got (1/20)^3, since it doesn't matter which symbol appears first, the other three just have to match the first symbol.

I am stuck on how to find pi (or probability of success) for b). I understand that n (number of trials) = 500 and x (number of successes) = 100.

The equation for binomials is P(X) = C(n, x)(pi^x)(1-pi)^n-x

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  • $\begingroup$ Is \$1 the prize for getting the same symbols on four reels? (Not a very generous payout!) If so, you get $n = 500$ plays for \$5. You need at least 1 win in 500, so $X \sim \mathsf{Binom}(n=500, p),$ with $p$ as in (a). You want $P(X \ge 1) = 1 - P(X = 0).$ I got about 0.0606. $\endgroup$ – BruceET Feb 22 at 23:39
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Call $\displaystyle p = {20\choose 1}20^{-4} = 20^{-3}$ the probability of success when playing once.

If you do this $n$, each repetition is independent. Hence, if $X$ is the number of times you won, then $X\sim\operatorname{Binom}(n, p)$ just as BruceET said. In particular, you have $n=4$ and been asked about $$\mathbb P(X\ge 1) = 1 - \mathbb P(X = 0) = 1 - (1 - p)^4.$$

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