A very good question. The answers given by other students are all correct, but I feel that they do not address the main confusion in the original question.

The problem is in the following assumption - quoting from the original posting "supposed that the error probability in passband and in baseband are the same so their SNR should be the same". This is not correct since you are not defining the term SNR in the same way for passband and baseband.

When you defined SNR in the passband as $E_{pass}/(N_0/2)$ you have taken the ratio of total signal energy (in all dimensions) to the noise variance *per dimension*. To illustrate the difference let us consider two examples and I think this will clarify things.

Case 1: QAM or PSK - In this case, there are two dimensions in the passband and only one complex dimension in the baseband. If you consider both dimensions together in the passband, the total signal energy is $E_{pass}$ and the total noise variance is $2 \times N_0/2 = N_0$. If you were to define SNR as the ratio of the total signal energy to the total noise variance then this would be $E_{pass}/N_0$ and not $E_{pass}/(N_0/2)$. So it will match with the definition of SNR in the baseband. Alternatively, you can keep the definition of passband SNR as signal energy in all dimensions/noise variance per dimension and redefine the SNR in baseband as signal energy/noise variance per dimension (real or imaginary component). Now both these quantities will be $2E_{pass}/N_0$

The confusion probably comes because of PAM and so I will discuss this now.

Case 2: PAM - In this case, there is only dimension in the passband and the baseband. So the SNR in the passband can be defined as $E_{pass}/(N_0/2)$. But the main point is that even though the noise in the baseband is complex and has variance $N_0$ in the real and imaginary components, the probability of error depends *only* on the real part of the received vector and hence the variance of the imaginary part of the noise component is not relevant for the probability of error. So by defining SNR to be $2E_{pass}/(2N_0)$, we are really using a definition of SNR that is not relevant to the probability of error. We should really look only at the real component since the signal is present only in the real component! Thus, we should define SNR as $2E_{pass}/N_0$ which is the same as that for passband!

The bottom line is that the term SNR does not have a universal definition and is one of the most abused terms in digital communications. This causes a lot of confusion. We can define SNR however you want but you should be careful that error probability expressions will change accordingly.

This is why I prefer to use $E_{s}/N_0$ where $E_s$ is the energy of the transmitted signals. If we have passband signals then $E_s$ is the energy in the passband signals and $N_0/2$ is the two-sided power spectral density of bandlimited passband noise. Now there is nothing ambiguous about this quantity $E_s/N_0$