Answer :
Let's address the problem step-by-step to determine whether the events [tex]\( A \)[/tex] (Edward purchases a video game) and [tex]\( B \)[/tex] (a store) are independent or dependent, and to identify the correct statement among the provided options.
Given:
- The probability that Edward purchases a video game from a store is [tex]\( P(A) = 0.67 \)[/tex].
- The probability that Edward purchases a game from a store is [tex]\( P(B) = 0.74 \)[/tex].
### Step-by-Step Solution:
1. Understanding Independence and Dependence:
- Two events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent if knowing that one occurs does not change the probability of the other occurring.
- Mathematically, [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent if [tex]\( P(A \mid B) = P(A) \)[/tex].
- Conversely, [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are dependent if [tex]\( P(A \mid B) \neq P(A) \)[/tex].
2. Calculating [tex]\( P(A \mid B) \)[/tex]:
- Here, [tex]\( P(A \mid B) \)[/tex] refers to the conditional probability that Edward purchases a video game given that he purchases from a store.
- We do not have additional specific information about [tex]\( P(A \mid B) \)[/tex] directly from the problem statement.
3. Analysis Based on Given Probabilities:
- With the given probabilities [tex]\( P(A) = 0.67 \)[/tex] and [tex]\( P(B) = 0.74 \)[/tex], and without additional information, the typical approach is to test the condition of independence.
- Hence, we assume [tex]\( P(A \mid B) = P(A) \)[/tex] in the absence of contrary information.
4. Testing Independence:
- If we assume [tex]\( P(A \mid B) = P(A) \)[/tex], then the events are independent according to the definition of independence.
5. Evaluating the Statements:
- Statement A: Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent because [tex]\( P(A \mid B) = P(A) \)[/tex].
- This correctly matches our assumption and definition of independence.
- Statement B: Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are dependent because [tex]\( P(A \mid B) = P(A) \)[/tex].
- This is contradictory because if [tex]\( P(A \mid B) = P(A) \)[/tex], the events should be independent, not dependent.
- Statement C: Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent because [tex]\( P(A \mid B) = P(B) \)[/tex].
- This is not true since [tex]\( P(A \mid B) \)[/tex] should equal [tex]\( P(A) \)[/tex] for independence, not [tex]\( P(B) \)[/tex].
- Statement D: Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are dependent because [tex]\( P(A \mid B) \neq P(A) \)[/tex].
- This statement is untrue under the given assumption [tex]\( P(A \mid B) = P(A) \)[/tex], implying independence.
Thus, the correct statement is:
A. Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent because [tex]\( P(A \mid B)=P(A) \)[/tex].
This confirms the independence of the events as per the given conditions and aligns with our logical approach.
Given:
- The probability that Edward purchases a video game from a store is [tex]\( P(A) = 0.67 \)[/tex].
- The probability that Edward purchases a game from a store is [tex]\( P(B) = 0.74 \)[/tex].
### Step-by-Step Solution:
1. Understanding Independence and Dependence:
- Two events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent if knowing that one occurs does not change the probability of the other occurring.
- Mathematically, [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent if [tex]\( P(A \mid B) = P(A) \)[/tex].
- Conversely, [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are dependent if [tex]\( P(A \mid B) \neq P(A) \)[/tex].
2. Calculating [tex]\( P(A \mid B) \)[/tex]:
- Here, [tex]\( P(A \mid B) \)[/tex] refers to the conditional probability that Edward purchases a video game given that he purchases from a store.
- We do not have additional specific information about [tex]\( P(A \mid B) \)[/tex] directly from the problem statement.
3. Analysis Based on Given Probabilities:
- With the given probabilities [tex]\( P(A) = 0.67 \)[/tex] and [tex]\( P(B) = 0.74 \)[/tex], and without additional information, the typical approach is to test the condition of independence.
- Hence, we assume [tex]\( P(A \mid B) = P(A) \)[/tex] in the absence of contrary information.
4. Testing Independence:
- If we assume [tex]\( P(A \mid B) = P(A) \)[/tex], then the events are independent according to the definition of independence.
5. Evaluating the Statements:
- Statement A: Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent because [tex]\( P(A \mid B) = P(A) \)[/tex].
- This correctly matches our assumption and definition of independence.
- Statement B: Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are dependent because [tex]\( P(A \mid B) = P(A) \)[/tex].
- This is contradictory because if [tex]\( P(A \mid B) = P(A) \)[/tex], the events should be independent, not dependent.
- Statement C: Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent because [tex]\( P(A \mid B) = P(B) \)[/tex].
- This is not true since [tex]\( P(A \mid B) \)[/tex] should equal [tex]\( P(A) \)[/tex] for independence, not [tex]\( P(B) \)[/tex].
- Statement D: Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are dependent because [tex]\( P(A \mid B) \neq P(A) \)[/tex].
- This statement is untrue under the given assumption [tex]\( P(A \mid B) = P(A) \)[/tex], implying independence.
Thus, the correct statement is:
A. Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent because [tex]\( P(A \mid B)=P(A) \)[/tex].
This confirms the independence of the events as per the given conditions and aligns with our logical approach.
