A question in Probability, risk of lung illness I was reading this book about medicine and I found this paragraph in it > A person's daily risk of getting lung illness ... is $0.5$ percent. The daily risk of getting a bad lung illness ... plus treatment is $0.05$ percent. So, you are looking at the difference between a $99.5$ percent chance of staying well, and a $99.95$ percent chance of staying well ... But sum it up over a year and it is the difference between an $83$ percent chance of making it through the year healthy, and a $16$ percent chance. I don't understand how this conclusion was reached. Was some additional data assumed?

No, they just assumed each day is independent. Without treatment, with $0.5$ percent chance per day, the chance of getting through a year is $.995^{365}=.160$ With treatment, you have $.9995^{365}=.833$.