Two dice are thrown 5 times, and each time sum of numbers obtained being 5 is consider

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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

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