Final answer:
Juror Three is likely embarrassed after his outburst about his personal life because it exposes his vulnerability and could potentially question his impartiality as a juror in a legal case.
Step-by-step explanation:
In Act One of the play, Juror Three experiences a moment of vulnerability when he has an outburst about his own child and their strained relationship. After this emotional revelation, the stage directions indicate that he is embarrassed, likely because he has unintentionally exposed his pain and weaknesses in front of others. This display of strong emotion, especially in a room full of strangers focused on a legal case, might make him feel that he has lost face or revealed too much of his private life, which is unrelated to the case at hand.
Moreover, this lapse into the personal may also bring into question his impartiality as a juror, making him feel uncomfortable about his professional conduct. Juror Three's embarrassment could also stem from the realization that his issues may be clouding his judgment, leading to concerns that he may not be living up to the civic responsibility of objectivity entrusted to him as a member of the jury.