Suppose I have two types of light bulbs, type-ex and type-wye, whose lifetimes in years are given by exponential random variables \(X\) and \(Y\), such that \(\mathbb{P}(X >x) = e^{-x/2}\) for all \(x \geq 0\), and \(\mathbb{P}(Y >y) = e^{-y/3}\) for all \(y \geq 0\).
Suppose I install one type-ex light bulb and one type-wye light bulb, and the light bulbs function independently of one another. What is the probability that the ex light bulb dies before the wye light bulb?
Suppose I flip a fair coin 20 times. Assume that the coin flips are independent. Show that the probability that I get a consecutive run of at least four heads is greater than \(0.27\). Hint: group the coins into 5 blocks.
Suppose that when Tessa buys a Kinder surprise candy egg to get the Kinder surprise toy. Each egg contains one of \(n\) toys uniformly at random, independently of each other. She wishes to collect all \(n\) toys. Let \(T\) be the number of toys her father has to buy in order for her to collect all \(n\) toys. Compute \(\mathbb{E} T\). You should obtain a nice formula that will allow you to compute easily \(\mathbb{E} T\) if \(n=7\). Hint: let \(T_i\) be the number of toys that need to be purchased, after collecting \(i-1\) distinct toys, then \[ T = T_1 + \cdots +T_n.\]