A popular problem by Brilliant Member

A Brilliant member named Billy Bob Joe posts \(n\) problems, where \(n\) is a positive integer. The probability that exactly three of those problems will become popular is \(n\) times the probability that exactly two of those problems will become popular. As \(n\) gets larger and larger, the probability that a problem posted by Billy Bob Joe will become popular approaches \(\frac{a}{b},\) where \(a\) and \(b\) are coprime positive integers. Find \(a + b.\)

Details and Assumptions:

  • Assume that the probabilities of each problem becoming popular are equal and independent of each other.
  • A problem can only be either popular or not popular. In other words, the probability that a problem will become either popular or not popular is equal to 1.
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