The sum of all probabilities must equal 1:

\(4p + 5p^2 + 1.5p + 2.5p + 1.5p = 1\)

Simplify the equation:

\(10p^2 + 19p = 1\)

Rearrange to form a quadratic equation:

\(10p^2 + 19p - 1 = 0\)

Using the quadratic formula \(p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 10\), \(b = 19\), and \(c = -1\):

\(p = \frac{-19 \pm \sqrt{19^2 - 4 \times 10 \times (-1)}}{2 \times 10}\)

\(p = \frac{-19 \pm \sqrt{361 + 40}}{20}\)

\(p = \frac{-19 \pm \sqrt{401}}{20}\)

Calculating the roots, we find \(p = 0.1\) or \(p = -2\).

Since probability cannot be negative, \(p = 0.1\).