Explanation
Solution
1. Condition for No Real Roots:
The quadratic equation 64x2+5Nx+1=0 has no real roots if the discriminant D < 0:
D = (5N)^2 - 4(64)(1) < 0 \text{}
25N^2 - 256 < 0 \implies N^2 < \frac{256}{25} \text{}
Since N must be a positive integer (number of tosses), the possible values for N are 1, 2, and 3.
2. Calculate the Probability:
N follows a Geometric Distribution where P(Head)=41 and P(Tail)=43.
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P(N=1)=41
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P(N=2)=(43)(41)=163
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P(N=3)=(43)2(41)=649
Total probability qp:
qp=41+163+649=6416+12+9=6437
Here, p=37 and q=64 (which are co-prime).
3. Final Calculation:
Final Answer: 27
