Step 1: Name the pass probabilities.
Let the chances that Amar, Akbar, Anthony pass be $a$, $k$, $t$. Since they work independently, joint probabilities multiply.
Step 2: Turn the three given facts into equations.
Amar passes and Akbar fails: $a(1 - k) = \frac{3}{20}$. Akbar passes and Anthony fails: $k(1 - t) = \frac{1}{4}$. Amar and Anthony both pass: $a t = \frac{2}{5}$.
Step 3: Guess friendly values that fit.
Try $a = \frac{3}{5}$, $k = \frac{3}{4}$, $t = \frac{2}{3}$. Check the first: $\frac{3}{5}\left(1 - \frac{3}{4}\right) = \frac{3}{5}\cdot\frac{1}{4} = \frac{3}{20}$. Good.
Step 4: Verify the other two facts.
Second: $\frac{3}{4}\left(1 - \frac{2}{3}\right) = \frac{3}{4}\cdot\frac{1}{3} = \frac{1}{4}$. Good. Third: $\frac{3}{5}\cdot\frac{2}{3} = \frac{2}{5}$. Good. All three match, so these values are correct.
Step 5: Find the chance that all three pass.
$P(\text{all pass}) = a\,k\,t = \frac{3}{5}\cdot\frac{3}{4}\cdot\frac{2}{3} = \frac{18}{60} = \frac{3}{10}$.
Step 6: Take the complement for at least one failing.
At least one fails is the opposite of all passing: $1 - \frac{3}{10} = \frac{7}{10}$.
\[ \boxed{\tfrac{7}{10}} \]