The Road To Success

Calculus Level 4

If something has a 1 in $n$ chance of success, then after $n$ tries, we expect to have succeeded once.

Approximately, as $n$ tends to infinity, what is the limit of the probability that with a 1 in $n$ chance of success, that after $n$ tries we have at least 1 success?

\approx 100\%

\approx 20\%

\approx 40\%

\approx 60\%

\approx 80\%

Limit does not exist

\approx 0\%

1 solution

Brian Charlesworth
May 15, 2018

On any given try the probability of failure is $1 - \dfrac{1}{n}$ , so the probability of $n$ failures in a row is $P_{n} = \left(1 - \dfrac{1}{n}\right)^{n}$ , which tends to $e^{-1} = \dfrac{1}{e}$ as $n \to \infty$ .

Thus the probability $1 - P_{n}$ of at least one success after $n$ tries as $n \to \infty$ is $1 - \dfrac{1}{e} \approx 0.632$ , or $\boxed{\approx 60 \text{\%}}$ .

Mistakenly took the last step to be $1-e^{-n}$ , get wrong answer $100\%$ ... Good solution!

Kelvin Hong - 3 years ago

The Road To Success

1 solution

0 pending reports