Alice is $30$ years old and owns a $25-$year term insurance policy that pays $$1,000,000$ at the end of year of death with actuarial present value equal to $$50,000.$

Alice's twin brother, Bob, has the same type of insurance policy except that the benefit is only $$500,000$ for the first six months $($goes up to $$1,000,000$ from month seven$).$ Payment is still done at the end of the year even if he dies within the first six months. The actuarial present value of his insurance policy is $$45,000.$

Assuming uniform distribution of deaths between integer years and $i=5\%,$ which value is closest to $q_{30}?$

$0\mathrm{.}200$

$0\mathrm{.}033$

$0\mathrm{.}020$

$0\mathrm{.}021$

$0\mathrm{.}010$