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Binomial Distribution — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsBinomial Distribution3 Marks
Question
In binomial distribution with five Bernoulli’s trials, the probability of one and two success are 0.4096 and 0.2048 respectively. Find the probability of success.
✓

Answer

Given: $X \sim B(n=5, p)$
The probability of $X$ success is
$
P(X=x)={ }^n \mathrm{C}_x p^x q^{n-x}, x=0,1,2, \ldots, n
$
i.e. $P(X=x)={ }^5 \mathrm{C}_x p^x q^{5-x}, x=0,1,2,3,4,5$
Probabilities of one and two successes are
$
P(X=1)={ }^5 C_1 p^1 q^{5-1}
$
and $P(X=2)={ }^5 \mathrm{C}_2 p^2 q^{5-2}$ respectively
Given: $P(X=1)=0.4096$ and $P(X=2)=0.2048$
$\therefore \frac{P(X=2)}{P(X=1)}=\frac{0.2048}{0.4096}$
i.e. $\frac{{ }^5 \mathrm{C}_2 p^2 q^{5-2}}{{ }^5 \mathrm{C}_1 p^2 q^{5-1}}=\frac{1}{2}$
i.e. $2 \times{ }^5 C_2 p^2 q^3=1 \times{ }^5 C_1 p q^4$
i.e. $2 \times \frac{5 \times 4}{1 \times 2} \times p^2 q^3=1 \times 5 \times p q^4$
i.e. $20 p^2 q^3=5 p q^4$
i.e. $4 p=q$
i.e. $4 p=1-p$
i.e. $5 p=1$
$
\therefore p=\frac{1}{5}
$
Hence, the probability of success is $\frac{1}{5}$.

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