# Probability of any of five independent events occurring

I am trying to calculate the probability that any one of 5 independent events occur - each event has a 5% chance.

I calculated by saying the probability of any of n events is:

p1 = 0.05

pn = (1 - pn-1) * p1 + pn-1

For p5 I calculate 22.6%. I also verified this my writing small program which simulates this and found the results close to 22.6%. Is my calculation correct?

The example is from the first chapter of: The Signal and the noise - showing the risk of CDOs. The book says the probability is 20.4% but my calculation says it should be 22.6%.

• What do you mean by uncorrelated events? Independent events? In which case your answer is correct but could also be found as $1-(1-.05)^5$. Jan 31, 2017 at 9:44
• Yes the book seems to use uncorrelated and independent interchangeably. Will update the question to say independent. Jan 31, 2017 at 9:47

What I think they meant was the probability that exactly one event occurs. There are five ways for this to happen, and in each case one of the five events occurs and the others don't, so the probability of each scenario is $0.05\times0.95^4=0.0407$. If you multiply this by $5$ (five ways to get just one event), you get $0.2036$, which (with a little rounding off) corresponds to the answer from your book.