The probability that the persons P₁ and P₂ will die in a year are p and q respectively. The probability that at the end of the year only of them will be alive, is equal to

p + q – 2pq

Let ‘A’ denotes the event that 'P₁' dies in a year and ‘B’ denotes the event that 'P₂' dies in a year.

Thus, required probability

$=P(\bar{A} \cap B)+P(A \cap \bar{B})$

$=P(A)+P(B)-2(A \cap B)$

$=p+q-2 p q$