Decidedly, statistics is hard. Everything above is wrong. Let's label the twenty drives D0 through D19. There are 2^20 possible scenarios, which can be represented as a string of 20 bits:

• 00000000000000000000 = all 20 drives are working

• 00000000000000000001 = D19-D1 working, D0 failing

• 00000000000000000010 = D19-D2 working, D1 failing, D0 working

• 00000000000000000011 = D19-D2 working, D1-D0 failing

• etc

The probability of each of these scenarios is:

• 00000000000000000000: .97^20

• 00000000000000000001: .97^19 × .03

• 00000000000000000010: .97^19 × .03

• 00000000000000000011: .97^18 × .03^2

• etc

There are C(20,4) = 4845 scenarios with exactly four failing drives (four "1" bits.) The probability of each scenario is .97^16 × .03^4. Therefore the probability of 4 failing drives (any drive) is the sum of the probability of each scenario: .97^16 × .03^4 × C(20,4) like I said 3 comments above.

However the probability of a file being irrecoverable is P(4 failing drives) + P(5 failing drives) + ... + P(20 failing drives):

    .97^16 × .03^4 × C(20,4)
  + .97^15 × .03^5 × C(20,5)
  + ...
  + .97^0 × .03^20 × C(20,20)
  = 0.267%