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Chapter 2: Problem 22
If a fair coin is successively flipped, find the probability that a head first appears on the fifth trial.
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A point is uniformly distributed within the disk of radius 1 . That is, its density is $$ f(x, y)=C, \quad 0 \leqslant x^{2}+y^{2} \leqslant 1 $$ Find the probability that its distance from the origin is less than \(x, 0 \leqslant x \leqslant 1\).
A total of \(r\) keys are to be put, one at a time, in \(k\) boxes, with each key independently being put in box \(i\) with probability \(p_{i}, \sum_{i=1}^{k} p_{i}=1 .\) Each time a key is put in a nonempty box, we say that a collision occurs. Find the expected number of collisions.
An airline knows that 5 percent of the people making reservations on a certain flight will not show up. Consequently, their policy is to sell 52 tickets for a flight that can hold only 50 passengers. What is the probability that there will be a seat available for every passenger who shows up?
Suppose that \(X\) and \(Y\) are independent binomial random variables with parameters \((n, p)\) and \((m, p) .\) Argue probabilistically (no computations necessary) that \(X+Y\) is binomial with parameters \((n+m, p)\).
Show that $$ \lim _{n \rightarrow \infty} e^{-n} \sum_{k=0}^{n} \frac{n^{k}}{k !}=\frac{1}{2} $$ Hint: Let \(X_{n}\) be Poisson with mean \(n\). Use the central limit theorem to show that \(P\left\\{X_{n} \leqslant n\right\\} \rightarrow \frac{1}{2}\).
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