I throw darts repeatedly. Assume that on each throw I have a 1% chance of hitting the bullseye, independently of all other throws. (Note that this implies for example that repetition doesn’t help my aim get any better; in my case that might not be such a bad assumption.) Find the chance that it takes me more than 100 throws to hit the bullseye.
[Math] Probability of hitting the bullseye in darts
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Best Answer
You want to calculate the probability of $100$ failures in a row. Since the probability of failure on any trial is $0.99$, the probability of $100$ failures in a row is $(0.99)^{100}$.
Remark: This problem is more interesting than it looks. If we use a calculator, the probability turns out to be $\approx 0.3660323$
Let's look at a related problem, where the probability of success is $\frac{1}{n}$, and therefore the probability of failure is $1-\frac{1}{n}$. If we repeat the experiment $n$ times independently, the probability of $n$ failures in a rowis $\left(1-\frac{1}{n}\right)^n$.
For large $n$, this is approximately $e^{-1}\approx 0.3678794$. So the probability of $n$ failures in a row when the probability of success is $\frac{1}{n}$ is, or large $n$, essentially independent of $n$.
It is possible that you were given this question in the context of the Poisson distribution. The number of successes in $100$ independent trials has binomial distribution. With our particular numbers, the binomial distribution is well approximated by the Poisson with mean (parameter) $\lambda=(0.1)(100)=1$. So the probability of $0$ successes is approximately $e^{-\lambda}$, in this case $e^{-1}$.