# 20% in one year vs 4% per year over 5 years

I am looking for information for some extended family members.

A medical procedure has a 4% chance of stroke per year over 5 years. One family member has said that that is a 20% chance of stroke over all (5*4%). Others argue that it is less because 4% each year is smaller. I am not sure how to frame this question in order to get an accurate answer. Is the chance of something happening being 4% a year for 5 years the same as say an instantaneous 20% chance of something happening?

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## 1 Answer

If I interpret your question correctly, you want to know the probability of a stroke occurring in $5$ years given that there is a constant $4\%$ chance of stroke each year?

Using complementary probability is the best approach here. Since there is a $4\%$ chance of a stroke happening each year, there is a $96\%$ chance of a stroke not happening each year. Then there is a $(.96)^5\approx.8153$, or $81.53\%$ chance that no stroke happens in those $5$ years. This means that there is a $100\%-81.53\%=18.47\%$ chance of stroke happening in those $5$ years, not $20\%$.

Think about this also, if something has a $4\%$ chance of happening each year, does this mean there is a $25\cdot 4\%=100\%$ chance that it happens in $25$ years? It doesn't make very much intuitive sense that a relatively rare event is guaranteed to happen in a certain time span.

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Thanks tons, Stack Exchange sites are always quick and reliable in terms of answers. –  user6256 Jan 26 '11 at 4:37
you switched digits from .8153 to 83.15%. Chance would be about 18.5%. Despite that, I stand by my earlier comment. I appreciate the fast and good help. –  user6256 Jan 26 '11 at 4:45
@chrisfs Yes, when writing my answer, I switched the $1$ and $5$ in $.8153$ accidentally, I have since edited it. My apologies! It should be correct now. –  yunone Jan 26 '11 at 4:47