Question

A dice has n sides with numbers 1, 2, 3, 4 n marked on it. If the probability that a particular number appears at
least once when the dice is rolled thrice is 217/729, what is the probability that on rolling the dice thrice, a particular
number would appear exactly twice?​

Answer 1

Answer:

8/243

Step-by-step explanation:

p(at least once)=1-p(not even once)=1-((n-1)/n)^3=217/729

on solving we get n=9

now required probability= 3C2* 1/9*1/9*8/9=8/243

Answer 2

Given :  A dice has n sides with numbers 1, 2, 3, 4 n marked on it.

If the probability that a particular number appears at least once when the dice is rolled thrice is 217/729,  

To Find : the probability that on rolling the dice thrice, a particular

number would appear exactly twice?​

Solution:

P(x) = ᵃCₓpˣqᵃ⁻ˣ    a is number of throws

A dice has n sides with numbers 1, 2, 3, 4 n marked on it

Probability of a particular number to appear  p = 1/n

not to appear q  =  1 - 1/n  = (n - 1)/n

dice is rolled thrice

probability that a particular number appears at least once = 1 - probability at a particular number does not  appears at all

P(0) = ³C₀(1/n)⁰((n - 1)/n)³⁻⁰

1 - P(0)  =  217/729

1  -  (( n - 1)/n)³  = 217/729

=>  (( n - 1)/n)³  = 512/729

=> (n - 1)/n = 8/9

=>  9n - 9  = 8n

=> n = 9

 a particular number would appear exactly twice on rolling the dice thrice,

P(2) =  ³C₂(1/9)²(8/9)³⁻²

= 24/729

probability that on rolling the dice thrice, a particular number would appear exactly twice = 24/729

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