180180 students in a tenth grade high school class take a survey about which video game consoles they own. 808080 students answer that one of their consoles is a Playstation, 909090 answer that one of their consoles is an Xbox. Out of these, there are 303030 who have both systems.
Let AAA be the event that a randomly selected student in the class has a Playstation and BBB be the event that the student has an Xbox. Based on this information, answer the following questions.
What is P(A)P(A)P, left parenthesis, A, right parenthesis, the probability that a randomly selected student has a Playstation?
What is P(B)P(B)P, left parenthesis, B, right parenthesis, the probability that a randomly selected student has an Xbox?
What is P(A\text{ and }B)P(A and B)P, left parenthesis, A, start text, space, a, n, d, space, end text, B, right parenthesis, the probability that a randomly selected student has a Playstation and an Xbox?
What is P(B\text{ | }A)P(B | A)P, left parenthesis, B, start text, space, vertical bar, space, end text, A, right parenthesis, the conditional probability that a randomly selected student has an Xbox given that he or she has a Playstation?
Is P(B\text{ | }A)=P(B)P(B | A)=P(B)P, left parenthesis, B, start text, space, vertical bar, space, end text, A, right parenthesis, equals, P, left parenthesis, B, right parenthesis? Are the events AAA and BBB independent?