The sample space of picking a 2-character password using the letters Y, B, R, O, G, and P is:
BB, BR, BG, BP
RB, RR, RG, RP
GB, GR, GG, GP
PB, PR, PG, PP
If double letters are not allowed, then the sample space is reduced to:
BR, BG, BP, RB, RG, RP, GB, GR, GP, PB, PR, PG
The probability of choosing a password with no Y's is 10/12 or 5/6, since there are 10 passwords that do not contain Y and 12 possible passwords in total.
The probability of choosing a password with no O's is also 10/12 or 5/6, since there are 10 passwords that do not contain O and 12 possible passwords in total.
The probabilities are equal since there are the same number of passwords that do not contain Y and do not contain O.