91Ó°ÊÓ

A statistical experiment has 11 equally likely outcomes that are denoted by \(a, b, c, d, e, f, g, h, i, j\), and \(k .\) Consider three events: \(A=\\{b, d, e, j\\}, B=\\{a, c, f, j\\}\), and \(C=\\{c, g, k\\}\) a. Are events \(A\) and \(B\) independent events? What about events \(A\) and \(C ?\) b. Are events \(A\) and \(B\) mutually exclusive events? What about \(A\) and \(C\) ? What about \(B\) and \(C\) ? c. What are the complements of events \(A, B\), and \(C\), respectively, and what are their probabilities?

Define the following two events for two tosses of a coin: \(A=\) at least one head is obtained \(B=\) two tails are obtained a. Are \(A\) and \(B\) mutually exclusive events? Are they independent? Explain why or why not. b. Are \(A\) and \(B\) complementary events? If yes, first calculate the probability of \(B\) and then calculate the probability of \(A\) using the complementary event rule.

Let \(A\) be the event that a number less than 3 is obtained if you roll a die once. What is the probability of \(A ?\) What is the complementary event of \(A\), and what is its probability?

What is meant by the joint probability of two or more events? Give one example.

What is the joint probability of two mutually exclusive events? Give one example.

Given that \(A, B\), and \(C\) are three independent events, find their joint probability for the following. a. \(P(A)=.81, \quad P(B)=.49\), and \(P(C)=.36\) b. \(P(A)=.02, \quad P(B)=.03, \quad\) and \(\quad P(C)=.05\)

Given that \(P(B)=.29\) and \(P(A\) and \(B)=.24\), find \(P(A \mid B)\).

Given that \(P(A \mid B)=.44\) and \(P(A\) and \(B)=.33\), find \(P(B)\).

In a statistics class of 42 students, 28 have volunteered for community service in the past. If two students are selected at random from this class, what is the probability that both of them have volunteered for community service in the past?

How is the addition rule of probability for two mutually exclusive events different from the rule for two events that are not mutually exclusive?