$P (A) = \frac{1}{3}$ , $P (B ∣ A) = \frac{3}{4}$ and $P (B ∣ \overline{A}) = \frac{1}{6}$ . What is $P (B)$ ?

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Two independent events $A$ and $B$ have $P (A) = 0.3$ and $P (B) = 0.5$ . What is $P (A \cap B)$ ?
Suppose $P (A) = 0.4$ , $P (B ∣ A) = 0.5$ and $P (B ∣ \overline{A}) = 0.2$ . What is $P (B)$ ?
With the same data, $P (A) = 0.4$ , $P (B ∣ A) = 0.5$ , $P (B ∣ \overline{A}) = 0.2$ and $P (B) = 0.32$ , what is $P (A ∣ B)$ ?
$X$ follows a binomial distribution $B (4, 0.5)$ . What is $P (X = 2)$ ?
$X$ follows $B (10, 0.3)$ . What is the expected value $E (X)$ ?
$X$ follows $B (10, 0.3)$ . What is the variance $V (X)$ ?

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