Answer :
To determine the correct statement about the independence or dependence of events \(A\) and \(B\), let's use the definition of independent events in probability theory. Two events \(A\) and \(B\) are independent if and only if the probability of \(A\) occurring given that \(B\) has occurred is equal to the probability of \(A\) occurring on its own. Mathematically, this is represented as:
[tex]\[ P(A \mid B) = P(A) \][/tex]
Given:
- \(P(A) = 0.67\) (the probability that Edward purchases a video game)
- \(P(B) = 0.74\) (the probability that Greg purchases a video game)
- \(P(A \mid B) = 0.67\) (the probability that Edward purchases a video game given that Greg has purchased one)
To check for independence, compare \(P(A \mid B)\) and \(P(A)\):
[tex]\[ P(A \mid B) = 0.67 \][/tex]
[tex]\[ P(A) = 0.67 \][/tex]
We see that:
[tex]\[ P(A \mid B) = P(A) \][/tex]
Since \( P(A \mid B) \) is equal to \( P(A) \), it indicates that the occurrence of event \(B\) (Greg purchasing a video game) does not affect the probability of event \(A\) (Edward purchasing a video game).
Therefore, the correct statement is:
A. Events \(A\) and \(B\) are independent because \(P(A \mid B) = P(A)\).
So, the answer is:
A. Events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent because [tex]\(P(A \mid B) = P(A)\)[/tex]
[tex]\[ P(A \mid B) = P(A) \][/tex]
Given:
- \(P(A) = 0.67\) (the probability that Edward purchases a video game)
- \(P(B) = 0.74\) (the probability that Greg purchases a video game)
- \(P(A \mid B) = 0.67\) (the probability that Edward purchases a video game given that Greg has purchased one)
To check for independence, compare \(P(A \mid B)\) and \(P(A)\):
[tex]\[ P(A \mid B) = 0.67 \][/tex]
[tex]\[ P(A) = 0.67 \][/tex]
We see that:
[tex]\[ P(A \mid B) = P(A) \][/tex]
Since \( P(A \mid B) \) is equal to \( P(A) \), it indicates that the occurrence of event \(B\) (Greg purchasing a video game) does not affect the probability of event \(A\) (Edward purchasing a video game).
Therefore, the correct statement is:
A. Events \(A\) and \(B\) are independent because \(P(A \mid B) = P(A)\).
So, the answer is:
A. Events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent because [tex]\(P(A \mid B) = P(A)\)[/tex]
To determine if they are independent or dependent, we need to check if the probability of Edward purchasing a video game given that Greg has purchased a video game is equal to the
probability of Edward purchasing a video game regardless of Greg's purchase.
Given:
- |( P(E) = 0.67 |) (Probability that
Edward purchases a video game)
- (P(G) = 0.74 V (Probability that
Greg purchases a video game)
- ( P(EIG) = 0.67 \) (Probability that
Edward purchases a video game given that Greg has purchased a video game)
For events |(E l) and |( G () to be independent, the following must hold true:
IL P(EIG) = P(E) |J
From the given information:
I P(E|G) = 0.67 J
IL P(E) = 0.67 \]
Since |( P(EIG) = P(E) \), events | E
() and \(G |) are independent.
Therefore, the correct statement is:
C. Events (E) and |(G )) are independent because |( P(E|G)
probability of Edward purchasing a video game regardless of Greg's purchase.
Given:
- |( P(E) = 0.67 |) (Probability that
Edward purchases a video game)
- (P(G) = 0.74 V (Probability that
Greg purchases a video game)
- ( P(EIG) = 0.67 \) (Probability that
Edward purchases a video game given that Greg has purchased a video game)
For events |(E l) and |( G () to be independent, the following must hold true:
IL P(EIG) = P(E) |J
From the given information:
I P(E|G) = 0.67 J
IL P(E) = 0.67 \]
Since |( P(EIG) = P(E) \), events | E
() and \(G |) are independent.
Therefore, the correct statement is:
C. Events (E) and |(G )) are independent because |( P(E|G)
