# Bayesian inference

I'm a bit confused with arranging the Bayes equation to update probability. Say, I have the following data:

$P(\text{blue birds in the whole study area}) = 0.16$; $P(\text{all except blue colored birds in the whole study area}) = 0.84$; $P(\text{birds in NW of the study area is blue}) = 0.22$; and $P(\text{blue birds outside the NW part of the study area}) = 0.11$;

Is that correct if I write:

$P(\text{blue|NW}) = \frac{P(\text{blue}) \cdot P(\text{NW|blue})}{P(\text{blue}) \cdot P(\text{NW|blue}) + P(\neg \text{blue}) \cdot P(\text{blue|}\neg\text{NW})} = \frac{0.16 \cdot 0.22}{0.16 \cdot 0.22 + 0.84 \cdot 0.1} = 0.28$?

Therefore, the probability of finding a blue bird in the NW of the study site has increased from the prior estimate of $16%$ to $28%$.

The confusing part is that we also know: $P(\text{birds in NW that are not blue}) = 0.78$ and if I use this information in the equation as $P(\neg\text{blue|NW})$, then the calculation stands as: $0.16*0.22/(0.16*0.22 + 0.84*0.78) = 0.054$ or $5.4\%$ only (though the probability of finding a blue bird in the NW is supposed to increase)?!?

In sum, which one is correct to use: $P(\text{blue|}\neg\text{NW})$ or $P(\neg\text{blue|NW})$ in this particular case or the whole idea is wrong??

Thanks.