For 6 trials of an experiment, let X be a binomial variate which satisfies the relation \( 9P(X=4)=P(X=2) \), then the probability of success is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \displaystyle\frac{1}{4} \)
For a binomial variate with n = 6, \( P(X=r)={}^{6}C_{r}\,p^{r}q^{6-r} \), where \( q=1-p \). The given relation is
\( \displaystyle 9\cdot{}^{6}C_{4}\,p^{4}q^{2}={}^{6}C_{2}\,p^{2}q^{4} \)
Since \( {}^{6}C_{4}={}^{6}C_{2} \), cancelling \( {}^{6}C_{2}\,p^{2}q^{2} \) gives
\( \displaystyle 9p^{2}=q^{2} \)
Both p and q are positive, so taking square roots,
\( \displaystyle 3p=q \)
\( \displaystyle 3p=1-p \)
\( \displaystyle p=\frac{1}{4} \)