Let the probability of getting head for a biased coin be . It is tossed repeatedly until a head appears. Let be the number of tosses required. If the probability that the equation has no real root is , where and are co-prime, then is equal to ______.

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Solution

Q.
Let the probability of getting head for a biased coin be $\frac{1}{4}$ . It is tossed repeatedly until a head appears. Let $N$ be the number of tosses required. If the probability that the equation $64 x^{2} + 5 N x + 1 = 0$ has no real root is $\frac{p}{q}$ , where $p$ and $q$ are co-prime, then $q - p$ is equal to ______. [2023]

Ans.
(27)

Let $H$ be the event of getting head and $T$ be the event of getting tail. $P (H) = \frac{1}{4} and P (T) = \frac{3}{4}$

Now, $64 x^{2} + 5 N x + 1 = 0$

$D = 25 N^{2} - 4 (64) (1) = 25 N^{2} - 256$

For no real roots, $D < 0$

$\Rightarrow 25 N^{2} - 256 < 0$

$\Rightarrow 25 N^{2} < 256 \Rightarrow N^{2} < \frac{256}{25} \Rightarrow N < \frac{16}{5}$

$∴$ $N = 1, 2, 3$

Required probability $= H + T H + T T H$

$= \frac{1}{4} + \frac{3}{4} \times \frac{1}{4} + \frac{3}{4} \times \frac{3}{4} \times \frac{1}{4}$

$= \frac{1}{4} + \frac{3}{16} + \frac{9}{64} = \frac{37}{64} = \frac{p}{q} (Given)$

$∴$ $q - p = 64 - 37 = 27$

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