Probability of receiving a job offer on 3rd application

Let $a = 0.1$ the probability to get an interview and $b = 0.2 = 2a$ the probability to be hired after an interview.

The probability to be hired at a given submission is $ab$.

The probability you are looking for is:

$$p = \underbrace{ab}_{\text{1st sub. hired}} + \underbrace{(1-ab)}_{\text{1st sub. fail}}\cdot\left(\underbrace{ab}_{\text{2nd sub. hired}} + \underbrace{(1-ab)}_{\text{2nd sub. fail}}\cdot\underbrace{ab}_{\text{3rd sub. hired}}\right) =\\ = a^3b^3 - 3a^2b^2 + 3ab = \\ = a^3 (2a)^3 - 3a^2 (2a)^2 + 3a(2a) = \\ = 8 a^6 - 12 a^4 + 6 a^2 = \\ = 8 \cdot 10^{-6} - 12 \cdot 10^{-4} + 6 \cdot 10^{2} = 0.058808.$$