# Probability problem, independent events

If there is a 5 percent chance of getting a particular result, and there are 15 of these events taking place simultaneously with the exact same probability, then collectively, what is the chance that one of these events will take place?

For example, if 15 cars of the same make and model, identical in terms of manufacturing, have a 5 percent chance of developing a fuel injection problem within 5 years.. well, what are the chances that one of these 15 cars will have a problem within 5 years? I'm quite sure you dont just add the probabilities up. Any help with this problem would be most welcome.

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haruspex
Homework Helper
Gold Member
If n independent events each have probability p of occurring then the probability that exactly r occur is given by the binomial distribution:
$^{n}C_{r}p^{r}(1-p)^{n-r}$
where $^{n}C_{r} = \frac{n!}{r!(n-r)!}$
The case you ask about is p = .05, n = 15, r = 1.
But perhaps you meant at least one occurring. In that case we have
$1-^{n}C_{r}p^{r}(1-p)^{n-r}$
where r = 0. I.e., 1-(1-p)n