Calculating the Probability of a Subset

Question

This is my first time facing a topic in stochastics.

My Problem: i have a total set of 7000 Numbers and a fixed subset containing 450 Numbers. So let's assume Numbers from 1-450 are red, Numbers from 451-7000 are blue. So now my question is: If i guess 3 random numbers (numbers cannot be the same) what is the probability that I guess a red number? Guessing correct would be p = 450/7000, right? So trying this 3 times should give me the probability of 3 * 450/7000, but this would lead to 100% correctness at 16 tries, which seems not to be correct. I should only reach 100% correctness after 6551 tries.

Could anybody explain me how to calculate my problem? Maybe even explaining the formula to me?

Best Regards,

Korbi

EDIT: After further research and the help here I found an actual formula targeting my problem called Hypergeometric Distribution. This is exactly what i want to calculate and plot.

Thanks for your help

Answer 1

First we do the probability that you guess three blue numbers: $$ \frac{6550}{7000} $$ for the first number to be blue, then $$ \frac{6549}{6999} $$ for the second number to be blue, then $$ \frac{6548}{6998} $$ for the third number to be blue. Thus, $$ \frac{6550}{7000}\;\frac{6549}{6999}\;\frac{6548}{6998} $$ is the probability of getting three blue. The complement $$1 - \frac{6550}{7000}\;\frac{6549}{6999}\;\frac{6548}{6998} $$ is the probability of getting at least one red. Is that what you mean by "guessing red"?

Answer 2

Try to compute the probability of not guessing any red numbers first.