A and B appear for an interview for two posts the probability of A's selection is 1/3 and that of B's selection is 2/5. Find the probability that only one of them will be selected ?  

CONCEPT:

• If P(A) = x then P(A̅) = 1 - x 
• If A and B are two independent events then P(A ∩ B) = P(A) × P(B)

CALCULATION:

Given: A and B appear for an interview for two posts such that the probability of A's selection is \(\frac 13\) and that of B's selection is \(\frac 25\).

Let E = event that A is selected

Let F = event that B is selected

⇒ P(E) = \(\frac 13\) and P(F) = \(\frac 25\)

As we know that, if P(A) = x then P(A̅) = 1 - x 

⇒ P(E̅) = 1 - (\(\frac 13\)) = \(\frac 23\) and P(F̅) = 1 - (\(\frac 25\)) = \(\frac 35\)

∴ P(event that one of them is selected) = P(E ∩ F̅) + P(E̅ ∩ F)

First let's find out P(E ∩ F̅) and P(E̅ ∩ F)

As we know that, if A and B are two independent events then P(A ∩ B) = P(A) × P(B)

⇒ P(E ∩ F̅) = P(E) × P(F̅) = (\(\frac 13\)) × (\(\frac 35\)) = \(\frac 15\)

⇒ P(E̅ ∩ F) = P(E̅) × P(F) = (\(\frac 23\)) × (\(\frac 25\)) = \(\frac {4}{15}\)

⇒ P(event that one of them is selected) = (\(\frac 15\)) + (\(\frac {4}{15}\)) = \(\frac {7}{15}\)

Hence, the correct option is 2.