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  • A fair coin is tossed 12 times. If the probability that two heads do not occur consecutively is , then

A fair coin is tossed 12 times. If the probability that two heads do not occur consecutively is \mathrm{p}, then the value of \mathrm{4096 p} must be

Option: 1

376


Option: 2

375


Option: 3

377


Option: 4

330


Answers (1)

best_answer

Total number of outcomes =2 \times 2 \times 2 \ldots 12 times
=4096
Let \mathrm{ a_n} denote the number of outcomes in which two consecutive heads do not occur when the fair coin is tossed \mathrm{ n} times.

\mathrm{ \Rightarrow \quad a_1=2, a_2=3 }

for \mathrm{ n \geq 3 }, if the last outcome is \mathrm{ T }, then we can not have two consecutive heads in the first \mathrm{ (n-1) } tosses. This can happen in \mathrm{a_{n-1}} ways. If the last outcome is \mathrm{H}, we must have \mathrm{T\: \: the \: \: (n-1)^{\text {th }}} toss and we can not have two consecutive heads in the first \mathrm{(n-2)} tosses. This can happen in \mathrm{a_{n-2}} ways.

\mathrm{\Rightarrow a_n=a_{n-1}+a_{n-2} \text { for } n \geq 3 }

\mathrm{\Rightarrow a_{10}=144, a_{11}=233, }

\mathrm{\Rightarrow a_{12}=377 . }

Hence, the required probability is \mathrm{\frac{377}{4096}=p }

\mathrm{\therefore \quad 4096 p =4096 \times \frac{377}{4096} }

\mathrm{ =377 }



 

Posted by

vishal kumar

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