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Q.
If a fair coin is tossed $5$ times, the probability that heads does not occur two or more times in a row is
KVPYKVPY 2017
Solution:
A fair coin is tossed $5$ times.
Total number of sample space $= 2^{5}$
Probability of head does not occur two or more time in rows Possible cases Case I All tails $=\left(\frac{1}{2}\right)^{5}=\frac{1}{2^{5}}$ Case II $4$ tails and one head
$^{5}C_{4} \left(\frac{1}{2}\right)^{5}=\frac{5}{2^{5}}$ Case III $3$ tails and $2$ head $\times T \times T \times T \times $
$^{4}C_{2} \times\left(\frac{1}{2}\right)^{5}=\frac{6}{2^{5}}$ Case IV $2$ tails and $3$ heads $\times T \times T \times$
$^{3}C_{3} \times\left(\frac{1}{2}\right)^{5}=\frac{1}{2^{5}}$
$\therefore $ Required probability
$=\frac{1}{2^{5}}+\frac{5}{2^{5}}+\frac{6}{2^{5}}+\frac{1}{2^{5}}$
$=\frac{13}{2^{5}}$