Draw letters with replacement from the following box:
[S T A T I S T I C S]
Let $Y$ be the distribution of the number of times the letter 'A' is drawn until you get the letter 'S' 5 times. Find $P(Y = y)$, $E[Y]$ and $Var[Y]$.
I'm having trouble finding $P(Y = y)$. My initial thought was that I could use the Negative Binomial distribution, but I'm not looking for a number of trials until the rth success.
$Y$: the count of drawing 'A' until you draw the fifth 'S'
From first principles: The event $\{Y=y\}$ requires you to draw an arrangement of $y$ A, $4$ S, and some amount of the other letters, and then finally the last S.
In essence only draws of A or S are counted, and all draws of I,T,C are safely ignored. They are a distraction; so get rid of them.
What is the probability for drawing an arrangement of $y$ A and 4 S, then followed by fifth S, from [S,S,S,A] with replacement? It is: $~\mathsf P(Y=y)~$.
tl;dr:
The distribution is that of a Negative Binomial for counting successes before the fifth failure with success rate $1/4$.
