If the probability for $A$ to fail in an examination is 0.2 and that for $B$ is 0.3 , then the probability that either $A$ or $B$ fail is

Let $P(A)$ be the probability that $A$ fails and $P(B)$ be the probability that $B$ fails. Given $P(A) = 0.2$ and $P(B) = 0.3$.

Since the events of $A$ and $B$ failing are independent, the probability that both fail is $P(A \cap B) = P(A) \times P(B) = 0.2 \times 0.3 = 0.06$.

The probability that either $A$ or $B$ fails is given by the addition theorem of probability: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

Substituting the values, we get $P(A \cup B) = 0.2 + 0.3 - 0.06 = 0.5 - 0.06 = 0.44$.

Answer: 0.44