I like to think of Bayes' rule in terms of hypotheses and predictions. Take the hypotheses to be "is a woman" and "is not a woman"/"is a man", and the data to be "is over 6 foot tall". Our priors on the to hypotheses are roughly equal (50%). They predict the data with probability 1% (6' given a woman) and 7% (6' given a man). [Hence total probability of 6' is 1%50% + 7%50% = 4%.]
Now, our Woman hypothesis predicts the data more poorly than the Man hypothesis, so we need to update our confidence in the hypotheses appropriately. Bayes' rule tells us that this should be in proportion to how well they predicted the data. Originally we had Woman:Man odds as 1:1, so now it must be 1:7 (or 12.5% for Woman).
If you think of a column plot with hypotheses along the horizontal axis, where the width of each column is proportional to the prior, and the height is proportional to the strength of the prediction of the data. If you then "squash down" the columns, maintaining their area, but so that they all have the same height, the new widths give the posterior probability.
Now, our Woman hypothesis predicts the data more poorly than the Man hypothesis, so we need to update our confidence in the hypotheses appropriately. Bayes' rule tells us that this should be in proportion to how well they predicted the data. Originally we had Woman:Man odds as 1:1, so now it must be 1:7 (or 12.5% for Woman).
If you think of a column plot with hypotheses along the horizontal axis, where the width of each column is proportional to the prior, and the height is proportional to the strength of the prediction of the data. If you then "squash down" the columns, maintaining their area, but so that they all have the same height, the new widths give the posterior probability.