Let a biased coin have P(head)=41, tossed repeatedly until head appears. Let N be tosses required. If the probability the equation 64x2+5Nx+1=0 has no real root is qp (coprime), find q−p.
Answer: 27
Solutions · 1
Official solution
No real roots: D=25N2−256<0⇒N<516⇒N=1,2,3.
\[P = \frac14+\frac34\cdot\frac14+\frac34\cdot\frac34\cdot\frac14 = \frac{37}{64}\]
p=37,q=64⟹q−p=27.