91Ó°ÊÓ

Find the 25 th percentile of the distribution having pdf \(f(x)=|x| / 4\), where \(-2

We say that \(X\) has a Laplace distribution if its pdf is $$ f(t)=\frac{1}{2} e^{-|t|}, \quad-\infty

A person answers each of two multiple choice questions at random. If there are four possible choices on each question, what is the conditional probability that both answers are correct given that at least one is correct?

Players \(A\) and \(B\) play a sequence of independent games. Player \(A\) throws a die first and wins on a "six." If he fails, \(B\) throws and wins on a "five" or "six." If he fails, \(A\) throws and wins on a "four," "five," or "six." And so on. Find the probability of each player winning the sequence.

Let \(C_{1}, C_{2}, C_{3}\) be independent events with probabilities \(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}\), respectively. Compute \(P\left(C_{1} \cup C_{2} \cup C_{3}\right)\).

Let \(\mathcal{C}=R\), where \(R\) is the set of all real numbers. Let \(\mathcal{I}\) be the set of all open intervals in \(R\). The Borel \(\sigma\) -field on the real line is given by $$ \mathcal{B}_{0}=\cap\\{\mathcal{E}: \mathcal{I} \subset \mathcal{E} \text { and } \mathcal{E} \text { is a } \sigma \text { -field }\\} $$ By definition, \(\mathcal{B}_{0}\) contains the open intervals. Because \([a, \infty)=(-\infty, a)^{c}\) and \(\mathcal{B}_{0}\) is closed under complements, it contains all intervals of the form \([a, \infty)\), for \(a \in R\). Continue in this way and show that \(\mathcal{B}_{0}\) contains all the closed and half- open intervals of real numbers.

Let \(X\) have the pdf \(f(x)=4 x^{3}, 0

Let the three mutually independent events \(C_{1}, C_{2}\), and \(C_{3}\) be such that \(P\left(C_{1}\right)=P\left(C_{2}\right)=P\left(C_{3}\right)=\frac{1}{4} .\) Find \(P\left[\left(C_{1}^{c} \cap C_{2}^{c}\right) \cup C_{3}\right]\)

Each bag in a large box contains 25 tulip bulbs. It is known that \(60 \%\) of the bags contain bulbs for 5 red and 20 yellow tulips, while the remaining \(40 \%\) of the bags contain bulbs for 15 red and 10 yellow tulips. A bag is selected at random and a bulb taken at random from this bag is planted. (a) What is the probability that it will be a yellow tulip? (b) Given that it is yellow, what is the conditional probability it comes from a bag that contained 5 red and 20 yellow bulbs?

A bowl contains 10 chips numbered \(1,2, \ldots, 10\), respectively. Five chips are drawn at random, one at a time, and without replacement. What is the probability that two even-numbered chips are drawn and they occur on even- numbered draws?