Error bars usually indicate a Gaussian distribution, not a flat one. If you said P=0.4 +- 0.03, that indicates that your probability of the final probability estimate ending up outside the 0.3-0.5 range is less than a percent. This seems to meet your requirements.
If that doesn’t suffice, it seems that you need a full probability distribution, specifying the probability of every P-value.
Error bars usually indicate a Gaussian distribution, not a flat one. If you said P=0.4 +- 0.03, that indicates that your probability of the final probability estimate ending up outside the 0.3-0.5 range is less than a percent. This seems to meet your requirements.
If that doesn’t suffice, it seems that you need a full probability distribution, specifying the probability of every P-value.