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  • in a binomial distribution B(n, p=1/4), if possiblility of atleast one sucess is greater than or equal to 9/10, then n is greater than??

in a binomial distribution B(n, p=1/4), if possiblility of atleast one sucess is greater than or equal to 9/10, then n is greater than??

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\begin{array}{l} 0,1,2,3,4, \ldots ., n \\ P=1-\text { all fail } \\ S(P)=\frac{1}{4} \\ S(F)=1-S(P) \\ =1-\frac{1}{4} \\ =\frac{3}{4} \end{array}

\begin{array}{l} P=1-\left(\frac{3}{4}\right)^{n} \geq \frac{9}{10} \\ -\frac{9}{10} \geq\left(\frac{3}{4}\right)^{n} \\ \\ \frac{1}{10} \geq\left(\frac{3}{4}\right)^{n} \\ -1 \geq n \frac{\log _{10} 3}{4} \end{array}

\begin{array}{l} n \leq-\frac{1}{\frac{\log _{10} 3}{4}} \\ \leq-\frac{1}{\log _{10} 3-\log _{10} 4} \\ n \geq \frac{1}{\log _{10} 4-\log _{10} 3} \end{array}

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Satyajeet Kumar

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