This paper is devoted to the study of the fundamental solution to the Cauchy initial value problem for the Wright-Fisher equation \[ \partial_t u(x,t) = x(1-x)\partial^2_x u(x,t),\quad (x,t)\in (0,1)\times(0,\infty), \] with boundary conditions \[ \lim_{t\rightarrow 0} u(x,t) = \varphi(x),\quad x\in(0,1);\quad u(0,t) = u(1,t) = 0,\quad t\in(0,\infty). \] The authors use a stochastic approach and associate the Itô stochastic differential equation to the Wright-Fisher equation in the form \[ X(t,x) = x+\int_0^t\sqrt{2X(\tau,x)(1-X(\tau,x))}\,dB(\tau), \] proving that the fundamental solution of the Wright-Fisher equation is the density for the distribution of the solution to the Itô equation. Further results involve estimates on the fundamental solution to the Wright-Fisher equation as \(t\rightarrow 0\) and Taylor series expansions of the solutions.