I will answer this question as it applies in my jurisdiction (Utah).
First, no one has a right simply to choose your judge or choose to change the judge assigned to your case.
One way to get a change of judge is under Utah Rules of Civil Procedure, Rule 63A. Change of judge as a matter of right, “all parties joined in the action may, by unanimous agreement and without cause, change the judge assigned to the action by filing a notice of change of judge.” The process for doing so is articulated in more detail in Rule 63A.
Another way to disqualify your judge (and thus have a different judge assigned) is under Utah Code Rule 63. Disability or disqualification of a judge. Under Rule 63, a motion to disqualify a judge can be filed. The process for doing so is articulated in more detail in Rule 63.
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