1
$\begingroup$

If $x$ objects are randomly distributed to $n$ groups, what is the formula for working out how big $x$ needs to be for the probability that at least one of the groups gets an amount $y$ (or larger) to exceed $50\%$?

Specifically, I am interested in knowing how big $x$ needs to be if there are $n=7$ groups and I need there to be a $50\%$ probability at least one of the groups gets $y=30$ objects.

$\endgroup$
  • $\begingroup$ Do you mean "uniformly randomly distributed"? $\endgroup$ – EngrStudent Jul 27 '17 at 14:45
  • $\begingroup$ @Engr That's a fair assumption that universally is made when the groups are not otherwise distinguished from each other. Incidentally, $x=163$ objects are needed; $162$ won't quite do it. $\endgroup$ – whuber Jul 27 '17 at 14:46
0
$\begingroup$

You have $x$ objects and throw them randomly (with equal probabilities) into one of $n$ boxes. Another formulation is that you have a $n$-sided regular dice and throws it $x$ times. (I will use your notation even if it is somewhat unconventional for this case). The number of times the dice comes up $i$ is $X_i$, and then $X_1, X_2, \dotsc, X_n$ has a multinomial distribution. Define $$ M = \max_{i=1,\dotsc,n} X_i $$ and to solve your question we need to find the probability distribution of $M$. But that question is answered completely in Die 100 rolls no face appearing more than 20 times

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.