Let Events A and B be described as follows: P(A) = buying popcorn P(B) = watching a movie

Question

The probability that you watch a movie this weekend is 48% The probability of watching a movie this weekend and buying popcorn is 38%. If the probability of buying popcorn is 42%, are watching a movie and buying popcorn independent?

Yes, because P(B |A) = 0.90 and the P(A and B) = 0.38 are not equal.

Yes, because P(A | B) = 0.79 and the P(A) = 0.42 are not equal.

No, because P(A| B) = 0.79 and the P(A) = 0.42 are not equal.

No, because P(B | A) = 0.90 and the P(A) = 0.42 are not equal.

Answer 1

P(M)=.48

P(P) = .42

P(M^P) = .38

P(P/M)=P(P) if independent events,

P(M/P) = P(M) if independent

they are dependent if the occurrence of one event changes the probability of the other event

Bayes Theorem is an identity, always true,

P(M/P)P(P) = P(P/M)P(M)

or P(M/P) = P(P/M)P(M)/P(P)

but if they are independent, the Bayes Theorem "collapses to

P(M)P(P) = P(P)P(M), which is a more obvious identity, always true

P(M^P) = P(M)P(P) if they are independent events

If P(M)= .48 and P(P) = .42

then P(M^P) = (.48)(.42) = .2016

IF .P(M^P) = .38 the events are dependent since .38 does not equal .2016

If one event occurs, that increases the probability of the other event, making them dependent events

P(M)=.48=P(A)

P(P)=.42=P(B)

P(A/B) = .79

P(A/B)P(B)= .79(.42)=.3318

P(B/A) = .9

P(B/A)P(A)=.9(.48)= .432 = about .42 as in answer D) so C) is the best answer,

although none are fully accurate. Answer is No, an C is clearly the best answer