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Given [tex]\( P(A) = \frac{1}{3} \)[/tex] and [tex]\( P(B) = \frac{5}{12} \)[/tex], where [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent events, determine [tex]\( P(A \cap B) \)[/tex].

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To determine [tex]\( P(A \cap B) \)[/tex] for two independent events [tex]\( A \)[/tex] and [tex]\( B \)[/tex], we use the property of independent events which states that if two events are independent, the probability of both events occurring simultaneously is the product of their individual probabilities. That is:

[tex]\[ P(A \cap B) = P(A) \cdot P(B) \][/tex]

Given:
[tex]\[ P(A) = \frac{1}{3} \][/tex]
[tex]\[ P(B) = \frac{5}{12} \][/tex]

We will multiply these two probabilities:

[tex]\[ P(A \cap B) = \frac{1}{3} \times \frac{5}{12} \][/tex]

Multiplying the fractions, we get:

[tex]\[ P(A \cap B) = \frac{1 \times 5}{3 \times 12} = \frac{5}{36} \][/tex]

Thus, the probability [tex]\( P(A \cap B) \)[/tex] is:

[tex]\[ P(A \cap B) = 0.1388888888888889 \][/tex]

Therefore,

[tex]\[ P(A \cap B) \approx 0.1389 \][/tex]

This is the probability that both events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] occur simultaneously.
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