Probability
Biased Coin — Quadratic with No Real Roots
nta_pyq_2023_apr
Grade 12
Question:
A biased coin with $P(H)=\frac{1}{4}$ is tossed until head appears. If $N$ is the number of tosses and $P$(equation $64x^2+5Nx+1=0$ has no real roots) $=\dfrac{k}{?}$, then $k=27$.
Step-by-Step Solution
Key Concept: $P(N=r)=(\frac{3}{4})^{r-1}\cdot\frac{1}{4}$. Equation has no real roots when discriminant $<0$: $25N^2<256\Rightarrow N\leq3$. $P(N\leq3)$.
$k=27$.
Correct Answer: 27