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