The health risk associated with radon is not from the radon gas. It is inert and does not re-act electro-chemically with the environment. The gas in its pure form can be inhaled into the lungs and exhaled with no ill effect. The health risk is actually associated with the radon daughters.
This is because the beta decay does not have any mass. The alpha decay however is an actual piece of matter escaping from the nucleases of the atom and has an actual impact on adjacent tissue. Much like smoking it causes an irritation, which overtime as the aging process makes the lung tissue less resilient can affect the lungs ability to reproduce and replace itself.
The radon daughters also carry an electro chemical charge, which means that unlike the inert gas, they do have the ability to bond with dust particles in the air. Because of this they can be inhaled into the lungs. This means that they can possibly transform while they are in the lungs and release either alpha or beta radiation, which will impact on the lung tissue.
Look at the chart to the left and you can examine the radon decay chain in detail. Notice that each alpha decay changes the Atomic Weight while the beta decays changes only the atomic number.