If you're serving on a jury where the prosecution os, or might, seek the death penalty, then you must be willing to see that penalty imposed. The US is not supposed to have "jury nullification" which is where the jury thinks that the person committed the crime but acquits anyway because they don't think the person should be punished for it.
In the US each side has a number of challenges they can use to strike jurors for almost any reason (you can't strike based on race). But both sides can also get jurors struck for "cause" which means that there's something in their background which may make them biased. In criminal cases, for example, law enforcement officers will be struck because it's presumed that they will favor the police and prosecution. People who have been victims of similar types of crimes, or who have been accused of similar types of crimes, might also be struck, as would their close relatives. The conflicts are usually determined through questioNing of prospective jurors, either orally by the judge and attorneys or via written questionaires (the latter especially in large and more complex cases). Nowadays, in major trials, the attorneys might employ a jury consultant who specializes in trying to select favorable jurors.
In a death penalty case, they would ask any potential juror if they would be able to impose a death sentence should they find the person guuilty. If they say that they wouldn't then they get struck automatically. The jury has to be "death penalty qualified." One problem with this is that people who are willing to impose the death penalty are also more likely to convict someone. So a death penalty qualified jury is also one which is more likely to favor the prosecutions case. It also tends to demographically skew the jury since African Americans are less likely to support the death penalty than whites.
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