One way to check theory is to simulate the problem. The code snippet below (using Octave) gives the empirical value of $0.62980$ which is comparable to the answer you have: $0.62419$ (See computation at Wolfram Alpha).
Of course, simulations like the one below are no guarantee that your theoretical answer is correct. However, at the very least they can help rule out incorrect answers and serve as a secondary check to your theoretical analysis.
function ev = ProbEst() # Probability of hit is 1/5 probHit = 1/5; totalIter = 10000; noShots = 10; # Counter to track if the event of interest # has occurred eventOccurred = 0; # Set seed of random number generator rand("state",[11;22;3209;42038021;11]); for iter =1:totalIter targetHit = 0; for shot = 1:noShots if (rand(1,1) < probHit) targetHit += 1; endif endfor if (targetHit >= 2) eventOccurred += 1; endif endfor # Output empirical probability to console eventOccurred/totalIter endfunction