A coin minting machine randomly produces unbalanced coins so that the probability of getting a head in tossing a coin is a random variably $Y$. Supposed $Y$ has a pdf $f(y) = 2y$ for $0 <= y <= 1$ and $0$ otherwise. Randomly take one coin.
- Toss this coin, and let $X$ be 1 if the outcome is a head, and 0 if a tail. Find the probability $P(X=1)$.
- Toss the coin $n$ times. Find the probability of getting $k$ heads in the $n$ tosses, where $n$ and $k$ are positive integers, and $k <= n$.
- Toss this coin twice. If the first toss results in a tail, what is the conditional probability that the next toss is also a tail?
Hint for (3): use the Beta function
Edit: I've having trouble conceptualizing this question. For (1) Isn't P(X=1) equivalent to Y?
What are the strategies behind solving (2) and (3)? For 3) isn't the second coin toss independent of the first toss? Why is there a conditional probability?