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