Let $X$ and $Y$ be geometric random variables with parameter $p$, with $0 \leq p \leq 1$. Assume that $X$ and $Y$ are independent.
Let $n$ be an integer greater than $1$. Let $k$ be a natural number with $k\leq n$. Then prove the formula
\[P(X=k \mid X + Y = n) = \frac{1}{n-1}.\]
Add to solve later Alice tossed a fair coin until a head occurred. Then Bob tossed the coin until a head occurred. Suppose that the total number of tosses for Alice and Bob was $7$.
Assuming that each toss is independent of each other, what is the probability that Alice tossed the coin exactly three times?
Add to solve later A box of some snacks includes one of five toys. The chances of getting any of the toys are equally likely and independent of the previous results.
(a) Suppose that you buy the box until you complete all the five toys. Find the expected number of boxes that you need to buy.
(b) Find the variance and the standard deviation of the event in part (a).
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