If you have a population of 100 positive cases, 87 vaxed, 13 unvaxed. Let's assume the unvaxed death rate is 100% (it doesn't mater).

If the vaccine is 90% efficient, then 10% of the 87 vaxed will die, 100% of the 13 unvaxed will die, therefore the % of vaxed in the deaths will be (we are using the same formula) 10% * 87 / (10% * 87 + 13) = 40%

If the vaccine is 15% efficient, then 85% of the 87 will die, and the % of vaxed in the deaths will be 85% * 87 / (85% * 87 + 13) = 85%