Problem 25
Use the module from Exercise \(4.24\) to make a program for solving the problems below. 1\. What is the probability of getting two heads when flipping a coin five times? This probability corresponds to \(n=5\) events, where the success of an event means getting head, which has probability \(p=1 / 2\), and we look for \(x=2\) successes. 2\. What is the probability of getting four ones in a row when throwing a die? This probability corresponds to \(n=4\) events, success is getting one and has probability \(p=1 / 6\), and we look for \(x=4\) successful events. 3\. Suppose cross country skiers typically experience one ski break in one out of 120 competitions. Hence, the probability of breaking a ski can be set to \(p=1 / 120\). What is the probability \(b\) that a skier will experience a ski break during five competitions in a world championship? This question is a bit more demanding than the other two. We are looking for the probability of \(1,2,3,4\) or 5 ski breaks, so it is simpler to ask for the probability \(c\) of not breaking a ski, and then compute \(b=1-c\). Define "success" as breaking a ski. We then look for \(x=0\) successes out of \(n=5\) trials, with \(p=1 / 120\) for each trial. Compute \(b .\) Name of program file: binomial_problems.py.
Problem 26
Suppose that over a period of \(t_{m}\) time units, a particular uncertain event happens (on average) \(\nu t_{m}\) times. The probability that there will be \(x\) such events in a time period \(t\) is approximately given by the formula $$ P(x, t, \nu)=\frac{(\nu t)^{x}}{x !} e^{-\nu t} $$
