# Probability of getting something with a low probablity

If there are 100 marbles in a bag (1 red one, 99 green ones), then the probability of picking the red one is 1/100. But if I do 100 trials then I believe it is likely that I will pick the red one at least once in those 100 trials. I'm curious as to what this probability is; that is, what is the probability of picking the red marble if I run 100 trials.

NOTE: I replace the marble after I pick it so there are always 100 marbles in the bag.

I've always wondered about this question, because I've always believed that the odds are >50% to pick the red marble, but I'd like to know exactly what they are. This experiment can obviously be done with any other type of probability problem. I just used the marbles to illustrate one example.

• Good question,what have you tried yourself? – hhsaffar Nov 20 '13 at 18:11

The probability of getting green every time is $0.99^{100} = 0.36603...$ So the probability of not getting green every time (which is another way to say getting red at least once) is $1-0.366=0.634$.

When you have a large number of marbles and always pick as many times as there are marbles in all, the probability of getting any particular one of them at least once will converge to $1-e^{-1} = 0.63212...$. This is due to the well-known limit $$\lim_{n\to\infty} \left(1+\frac{x}{n}\right)^n = e^x$$ (which is sometimes used as a definition of $e^x$) applied at $x=-1$.

From the same limit: If you have $n$ marbles (and $n$ is large), the number of tries you need to do in order to have 50-50 chance of getting a selected one at least once is about $\ln(2) n \approx 0.693\,n$.

• Can you please explain briefly how it converges to $1-e^{-1}$? – hhsaffar Nov 20 '13 at 18:22
• @hhsaffar: done – Henning Makholm Nov 20 '13 at 18:27

Probability of at least one occurrence of the red marble equals 1- the probability of no occurrence of the red marble = $1-(0.99)^{100}$. That is around 63 percent.

• 0.01 is not the probability of "no red", but the probability of "red". – Henning Makholm Nov 20 '13 at 18:16
• Sorry, Thanks, you are right, I fixed it. – hhsaffar Nov 20 '13 at 18:17

The probability of getting a green marble is $0.99$. The odds of getting a green marble $100$ times in a row is $0.99^{100}$. Windows calculator gives me an answer of about $0.366$. So the probability of getting at least one red in $100$ tries is about $63.4$%.

Hint: What is the possibility that you pick out a green one? And what the probability that you do that each of the $100$ times?