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14.Probability
hard
Two dice are thrown $5$ times, and each time the sum of the numbers obtained being $5$ is considered a success. If the probability of having at least $4$ successes is $\frac{\mathrm{k}}{3^{11}}$, then $\mathrm{k}$ is equal to
A
$82$
B
$123$
C
$164$
D
$75$
(JEE MAIN-2023)
Solution
Probability of success $=\frac{1}{9}=p$
Probability of failure $q =\frac{8}{9}$
$P (\text { at least } 4 \text { success })= P (4 \text { success })+ P (5 \text { success })$
$={ }^5 C _4 p ^4 q +{ }^5 C _5 p ^5=\frac{41}{3^{10}}=\frac{123}{3^{11}}$
$k =123$
Standard 11
Mathematics
