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To determine whether events B and A are independent, we must check if the probability of their intersection [tex]\( P(B \cap A) \)[/tex] is equal to the product of their individual probabilities [tex]\( P(B) \times P(A) \)[/tex].
From the given table, we can extract the following probabilities:
- [tex]\( P(A) = 0.30 \)[/tex]
- [tex]\( P(B) = 0.40 \)[/tex]
- [tex]\( P(B \cap A) = 0.16 \)[/tex] (this value corresponds to the cell in the table where both B and A occur)
Now, let's calculate [tex]\( P(B) \times P(A) \)[/tex]:
[tex]\[ P(B) \times P(A) = 0.40 \times 0.30 = 0.12 \][/tex]
We then compare [tex]\( P(B \cap A) \)[/tex] and [tex]\( P(B) \times P(A) \)[/tex]:
- [tex]\( P(B \cap A) = 0.16 \)[/tex]
- [tex]\( P(B) \times P(A) = 0.12 \)[/tex]
Since [tex]\( P(B \cap A) \neq P(B) \times P(A) \)[/tex] (specifically, [tex]\( 0.16 \neq 0.12 \)[/tex]), events B and A are not independent.
Therefore, the correct answer is:
B. No, they are not independent because [tex]\( P(B \cap A) \neq P(B) \times P(A) \)[/tex]. [tex]\( P(B \cap A) = 0.16 \)[/tex] and [tex]\( P(B) \times P(A) = 0.12 \)[/tex].
From the given table, we can extract the following probabilities:
- [tex]\( P(A) = 0.30 \)[/tex]
- [tex]\( P(B) = 0.40 \)[/tex]
- [tex]\( P(B \cap A) = 0.16 \)[/tex] (this value corresponds to the cell in the table where both B and A occur)
Now, let's calculate [tex]\( P(B) \times P(A) \)[/tex]:
[tex]\[ P(B) \times P(A) = 0.40 \times 0.30 = 0.12 \][/tex]
We then compare [tex]\( P(B \cap A) \)[/tex] and [tex]\( P(B) \times P(A) \)[/tex]:
- [tex]\( P(B \cap A) = 0.16 \)[/tex]
- [tex]\( P(B) \times P(A) = 0.12 \)[/tex]
Since [tex]\( P(B \cap A) \neq P(B) \times P(A) \)[/tex] (specifically, [tex]\( 0.16 \neq 0.12 \)[/tex]), events B and A are not independent.
Therefore, the correct answer is:
B. No, they are not independent because [tex]\( P(B \cap A) \neq P(B) \times P(A) \)[/tex]. [tex]\( P(B \cap A) = 0.16 \)[/tex] and [tex]\( P(B) \times P(A) = 0.12 \)[/tex].
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