The amount of radiation we personally encounter that originates in the earth's and the atmosphere's natural background is?

More than half

Explanation:

The amount of radiation we personally encounter that originates in the Earth's and the atmosphere's natural background is more than half.

The reason is, since natural sources of background radiation include: cosmic rays, that is, radiation that reaches the Earth from space; rays from rocks and soil, that is, some rocks are radioactive and give off radioactive radon gas; rays from living things, like some plants absorb radioactive materials from the soil and these pass up the food chain.

And the main source of exposure for most people is naturally-occurring background radiation. Levels typically range from about 1.5 to 3.5 millisievert per year and can even be more than 50 mSv/yr.

From the above information therefore, the amount of radiation we personally encounter that originates in the Earth's and the atmosphere's natural background is more than half.