a person who is proud of being wrong

A contrarian is such a person:

= disagreed with by most people, or liking to express opinions that most people disagree with

Cambridge

From this and other definitions, if “most people” is taken to represent what is correct, it is clear that the contrarian likes to be wrong.

If we conjecture that “most people” may be wrong, the contrarian may prove to be right, but this is mere sophistry and speculation.

Willfully ignorant is a common expression used to describe such a person.

If you mean that the person knows that they are "wrong" about the "answer to a simple question", rather than that they just don't know, then there's:

mumpsimus: "a view or opinion stubbornly held, even when shown to be misguided; a person holding such a view, or one stubbornly adhering to old ways. [Chambers]

1. A person who obstinately adheres to old ways in spite of clear evidence that they are wrong; an ignorant and bigoted opponent of reform.

2. An obvious error that is obstinately repeated despite correction. Wiktionary

The story goes that a monk was in the habit of saying "mumpsimus" instead of "sumpsimus" when saying Mass, and insisted on saying "mumpsimus" even after being told the correct word.

Bold Ben has deleted his answer, which is "inverted intellectual snobbery" - This is a pity, as this works and what I would have suggested:

The character described in the example is delighting in being a member of a class that has no regard for achievement.

OED

Snob:

3d. A person who despises those whom he or she considers to be inferior in rank, attainment, or taste. Frequently in extended sense, with defining word limiting its reference to a particular sphere.

1911 G. B. Shaw Getting Married in Doctor's Dilemma 228 All her childish affectations of conscientious scruple and religious impulse have been applauded and deferred to until she has become an ethical snob of the first water.

1939 L. MacNeice Autumn Jrnl. xii. 49 Spiritually bankrupt Intellectual snobs.

and for "inverted"

(At Compounds - "inverted") inverted snobbery n. an attitude of exaggerated contempt or dislike for wealth, achievement, high social status, refined taste, typically combined with esteem for ‘ordinary people’, popular taste or culture, etc.; behaviour associated with such an attitude.

1937 L. Bromfield Rains Came i. xxxviii. 159 It was that eternal, inverted snobbery of his, that hatred of anyone born with the things he had never achieved.

1971 Lancet 23 Apr. 918/2 The curious inverted snobbery of the use of ‘Mr.’ by surgeons..is well worth the attention of a sociologist.