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10 cards from this deck
Game: win/lose, play up to & including first win
X∼Geo(p)X \sim \text{Geo}(p)X∼Geo(p), ppp=P(win), XXX=trials to 1st win
Trials are independent
Fixed probability of success
P(X=n)=(1−p)n−1pP(X=n)=(1-p)^{n-1}pP(X=n)=(1−p)n−1p
n−1n-1n−1 losses, then 1 win
(1−p)n(1-p)^n(1−p)n
P(X>n−1)P(X > n-1)P(X>n−1)
1−P(X>n)1 - P(X > n)1−P(X>n) or 1−(1−p)n1-(1-p)^n1−(1−p)n
(1−p)a−1−(1−p)b−1(1-p)^{a-1}-(1-p)^{b-1}(1−p)a−1−(1−p)b−1
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