It is possible to relate the joint life and last-survivor status without the assumption of independence. Consider the events (A =
{T(x) leq t}) and (B = {T(y) leq t}). Now,
begin{eqnarray*}
A textrm{ and } B &=& {T(x) leq t,T(y) leq t} = {T(overline{xy}) leq t}
end{eqnarray*} and
begin{eqnarray*}
A textrm{ or } B &=& {T(xy) leq t } .
end{eqnarray*} Because (Pr(A textrm{ or } B) = Pr(A)+Pr(B) – Pr(A textrm{ and } B)), we have
begin{eqnarray*}
Pr(T(xy) leq t) = Pr(T(x) leq t)+Pr(T(y) leq t) –
Pr(T(overline{xy}) leq t)
end{eqnarray*} so that
begin{eqnarray*}
~_t p_{overline{xy}} =~_t p_x + ~_t p_y – ~_t p_{xy}.
end{eqnarray*} Note that this relationship holds without using the assumption of independence between lives.