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For event A and event B, P(A) = 7/10, P(B) = 2/3, and P(A and B) 7/30. Are the events independent or dependent

Respuesta :

jacob193

Answer:

The two events are not independent.

Step-by-step explanation:

Two events [tex]A[/tex] and [tex]B[/tex] are independent if and only if the following are equal:

  • Probability that both [tex]A[/tex] and [tex]B[/tex] happen, [tex]P(A \cap B)[/tex], and
  • Product of the probabilities of the two events: [tex]P(A)\, P(B)[/tex].

In other words, one possible way to verify if [tex]A[/tex] and [tex]B[/tex] are independent is to check if [tex]P(A \cap B) = P(A)\, P(B)[/tex].

In this question, it is given that [tex]P(A \cap B) = (7/30)[/tex], [tex]P(A) = (7/10)[/tex], and [tex]P(B) = (2 / 3)[/tex].

  • [tex]\displaystyle P(A \cap B) = \frac{7}{30}[/tex].
  • [tex]\displaystyle P(A)\, P(B) = \frac{7}{10}\times \frac{2}{3} = \frac{14}{30}[/tex].

Because [tex]P(A \cap B) \ne P(A)\, P(B)[/tex], event [tex]A[/tex] and event [tex]B[/tex] are not independent.

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