Conditional Probability

due by Sep 17, 2018

All problems are from Feller.

1. Ten fair dice were thrown. Given that at least one of them produced one, what is the probability that two or more dice produced one.
2. In a bolt factory machines A, B, C manufacture 25, 35 and 40 percent of the total, respectively. Of their output 5, 4 and 2 per cent (respectively) are defective bolts. A bolt is drawn at random from the produce and is found defective. What are the probabilities that it was manufactured by machines A, B and C?
3. A (fair) die is thrown as long as necessary for one to turn up.
   1. Assuming that one does not turn up at the first throw, what is the probability that more than three throws will be necessary.
   2. Suppose that the number, $n$, of throws is even. What is the probability that $n=2$?
4. Die A has four red and two white faces, whereas die B has two red and four white faces. A coin is flipped once. If it falls head, die A is chosen and if it falls tail die B is chosen. The chosen die is thrown repeatedly.
   1. Show that the probability of red at any throw is $1/2$.
   2. If the first two throws result in red, what is the probability of red in the third throw?
   3. If red turns up at the first $n$ throws, what is the probability that die A is being used?
5. Let the probability that a family has exactly $n$ children be $p_n=\alpha p^n$ when $n\geq 1$ and $p_0 = 1 - \alpha p(1 + p + p^2 + \dots)$ for $n=0$. Suppose that all sex distributions of children have the same probability. Show that for $k\geq 1$ the probability that a family has exactly $k$ boys is $2\alpha p^k/(2 - p)^{k+1}$.