Solution. \begin {align*} P(X = r\, \big /\, X + Y = n) & = \frac {P(X = r\, , \, X + Y = n)}{P(X + Y = n)}\\\\ & = \frac {P(X = r\, , \, Y = n - r)}{P(X + Y = n)}\\\\ & = \frac {P(X = r)\, P(Y = n - r)}{P(X + Y = n)}\\\\ & = \frac {\dfrac {e^{-\lambda _1} \, \lambda _1^r}{r!}\, \, \dfrac {e^{-\lambda _2}\, \, \lambda _2^{n-r}}{(n - r)!}}{\dfrac {e^{-(\lambda _1 + \lambda _2)}\, \, (\lambda _1 + \lambda _2)^n}{n!}}\\ & = C^n_r\, \, \frac {\lambda _1^r\, \lambda _2^{n - r}}{(\lambda _1 + \lambda _2)^n}\\ & = C^n_r\, \, \left (\frac {\lambda _1}{\lambda _1 + \lambda _2}\right )^r\,\, \left (\frac {\lambda _2}{\lambda _1 + \lambda _2}\right )^{n - r}\hspace {0.3cm} r = 0\, , \, 1\, , \, \cdots \cdots , \, n. \end {align*} □