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You regularly attend baseball games. At these games, you have two methods for receiving free souvenirs: either by catching a foul ball or by catching a t-shirt thrown by the team mascot. Assume that your success at catching a foul ball is independent of your success at catching a t-shirt. Further, each game you have a probability $p_1$ of catching a foul ball and a probability $p_2$ of catching a t-shirt, independent of all the other games. Let random variable $X$ be the number of games you attend before catching at least one souvenir (e.g. if you got no souvenir the first game, but caught a foul ball the second game, then $X = 2$).

  1. What type of distribution does $X$ have, and what are its parameters?

  2. If $p_1 = 0.01$ and $p_2 = 0.02$, what is the expected number of games you must attend before you catch at least one souvenir?

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The probability of getting a souvenir at a game is $$ p=P(\text{ball}\vee\text{t-shirt})=1-(1-p_1)(1-p_2)=p_1+p_2-p_1p_2$$ The probability of getting the first souvenir at game $k$ is $$ P(X=k)=P(\text{nothing at $k-1$ games})P(\text{souvenir})=(1-p)^{k-1}p$$ This is called Negative binomial distribution $NB(1,1-p)$.

The mean ("expected number of games you must attend...") is $$E(X)=\sum_{n=1}^{\infty}n(1-p)^{n-1}p=\frac{1}{p}$$

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