How fast will this become popular?

Let $p$ be the probability that this problem becomes popular within 6 minutes of posting it. The value of $1 - p$ equals the probability that this problem does not become popular within 6 minutes. Since the probability of popularity remains the same throughout the half hour, we have that $(1 - p)^5$ equals the probability that this problem does not become popular within 30 minutes. We know that this value equals $\dfrac{3125}{16807}$ from the information we are given. Solving for $p$ , we get $p = \dfrac{2}{7}$ , so $a + b = 2 + 7 = \boxed{9}$ .

Half an hour is 5 blocks of 6 minutes each. As $\displaystyle \dfrac {13682}{16807} = 1 -\dfrac {3125}{16807} = 1 - \dfrac { 5^5}{ 7^5}$ . Hence the probability of not becoming popular is 5/7 and consequently that of becoming popular is 2/7. Answer: 9