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From the Wikipedia page of Alpha Ursae Majoris:

Alpha Ursae Majoris, Latinised from α Ursae Majoris, formally named Dubhe /ˈdʌbiː/, is, despite being designated "α" (alpha), the second-brightest object in the northern constellation of Ursa Major.

From the Wikipedia page of Epsilon Ursae Majoris:

Despite being designated "ε" (epsilon), it is the brightest star in the constellation and at magnitude 1.77 is the thirty-third brightest star in the sky.

Why are sometimes the brightest star of a constellation not named Alpha? How many constellations are there with this naming exception?

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Bayer didn't sort the stars strictly by brightness. He classified them as "first magnitude", "second magnitude" (and so on) following the traditional magnitudes assigned to stars since ancient times.

Then he names the stars of the first magnitude $\alpha$, $\beta$ etc. without attempting to distinguish the brightest star. Sometimes he seems to have labeled the stars from East to West, or in order of rising, or from the "head" to the "tail" of the constellation. Sometimes his ordering seems to have been haphazard.

And he didn't always follow his own "rules". In Ursa Major, the stars are named in sequence along the "dipper", not according to brightness.

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    $\begingroup$ Note that Ursa Major != the Big Dipper. The dipper is just part of the larger constellation of Ursa Major, which contains about 10 other stars in addition to those that make up the dipper. $\endgroup$
    – Darrel Hoffman
    Commented Jun 21, 2023 at 13:20
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It's true that stars are lettered in order of brightness. So, in a constellation the brightnest star is (Alpha), the second brightest (Beta) and so on. That is why most of the First Magnitude Stars are designated Alpha and a few are designated Beta. But there are cases where stars in a constellation have nearly the same brightness. In that case, they are lettered in order of their positions in the constellation, beginning at the head. For example, in the constellation of Ursa Major the bright stars are almost similar in brightness and therefore are lettered in order of postion from the head to the ladle.

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You can check Bayer designation for more information.

References

  1. http://www.stargazing.net/david/constel/greek.html

  2. https://skyandtelescope.org/astronomy-resources/the-greek-alphabet/

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