$P (A) = \frac{3}{10}$ , $P (B ∣ A) = \frac{1}{2}$ and $P (B ∣ \overset{ˉ}{A}) = \frac{1}{10}$ . What is $P (A ∣ B)$ ?

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A trial succeeds with probability $\frac{1}{4}$ . It is repeated 3 times independently. What is the probability of exactly 1 success?
A trial succeeds with probability $\frac{1}{4}$ . It is repeated 5 times independently. What is the probability of exactly 2 successes?
$P (A) = \frac{2}{5}$ , $P (B ∣ A) = \frac{1}{2}$ and $P (B ∣ \overset{ˉ}{A}) = \frac{1}{10}$ . What is $P (A ∣ B)$ ?
$P (A) = \frac{3}{10}$ , $P (B ∣ A) = \frac{1}{5}$ and $P (B ∣ \overset{ˉ}{A}) = \frac{2}{5}$ . What is $P (A ∣ B)$ ?
A trial succeeds with probability $\frac{1}{3}$ . It is repeated 5 times independently. What is the probability of exactly 2 successes?
$P (A) = \frac{1}{2}$ , $P (B ∣ A) = \frac{1}{2}$ and $P (B ∣ \overset{ˉ}{A}) = \frac{2}{5}$ . What is $P (A ∣ B)$ ?