public-notes/Choosing Stuff.md

#Math #Probability

Problem

Given m items of one type and n items of another type, what is the probability of choosing l items of type one and o items of type two if you pick l + o items?

Solution

Total ways to choose the items not considering types:

{m + n} \choose {l + o}

Total ways to choose l items of type one:

m \choose l

Total ways to choose o items of type two:

n \choose o

Multiply the ways to choose both items to get the number of ways to choose l items of type one and o items of type two, divide by total number of combinations:

\frac {{m \choose l} {n \choose o}} {{m + n} \choose {l + o}}