$\mathbb{P} P=5 D^{1} \cdot \frac{(1+i)^{2} i}{(1+1)^{N}-1}$

Step 1 ：Given values: \(D = 1\), \(i = 1\), and \(N = 5\)

Step 2 ：Plug these values into the formula: \(P = 5 \cdot 1^{1} \cdot \frac{(1+1)^{2} \cdot 1}{(1+1)^{5}-1}\)

Step 3 ：Simplify the expression: \(P = 5 \cdot \frac{2^{2}}{2^{5}-1}\)

Step 4 ：Calculate the value of P: \(P = 5 \cdot \frac{4}{31}\)

Step 5 ：\(\boxed{P \approx 0.6451612903225806}\)