I need help to understand how to work this problem. I am not expecting a full an

Question

I need help to understand how to work this problem. I am not expecting a full answer, although I'm not crazy if it were offered. I would just like to know how to figure this out.

Thank you in advance.

A genetic algorithm uses the following mutation operator: the bits in the input string are considered one by one independently, with probability 0.01 that each bit is inverted.

Given that you apply the mutate operator to the string (0 0 0 0), what is the probability that the output is:

(0 0 0 0)?

(0 1 0 0)?

(1 0 1 0)?

(1 1 1 1)?

Show the process of your computation.

Explanation / Answer

The chances of a bit not being flipped is 99.99% (.9999 in decimal) The chances of a bit being flipped is 0.01% = (0.0001 in decimal) (a) For (0 0 0 0), no bits were flipped. Therefore, the chances of this occurring is (.9999) * (.9999) * (.9999) * (.9999) = 0.9996000599960001 There is a 99.96% chance of this occurring. (b) For (0 1 0 0), 1 bit was flipped. Therefore, the chances of this occurring is (.9999) * (.0001) * (.9999) * (.9999) = 0.0000999700029999 There is a 0.0099% chance of this occurring. (c) For (1 0 1 0), 2 bits were flipped. Therefore, the chance of this occurring is (.0001) * (.9999) * (.0001) * (.9999) = 0.0000000099980001 There is a 0.0000009998% chance of this occurring. (d) For (1 1 1 1), 4 bits were flipped. Therefore, the chance of this occurring is (.0001) * (.0001) * (.0001) * (.0001) = 0.0000000000000001 There is a 0.00000000000001% chance of this occurring.

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