Probability of drawing a red ball if I draw 11 balls from an urn of 25 balls with 5 being red balls. Imagine you have an urn with $25$ balls in it. $20$ of the balls are black and $5$ of the balls are red. If I draw $11$ balls from the urn, what is the probability that none of them are red? I have written python code to test it, and empirically I believe it should be $\sim 1.85\text{%}$.
I had a naive idea to compute it as,
$$1 - \left((20/25)*(19/24)*...*(10/15)\right) \approx 0.024$$
I thought, the probability of getting a black ball on the first try is $20/25$, the probability of getting a black ball on the second, given I got one on the first, is $19/24$, and so on.
 A: First of all, there is error (probably a typo) in ''$\left((20/25)*(19/14)*...*(10/15)\right)$''. Secondly you shouldn't have subtracted it from $1$.
The probability changes from one draw to the next when balls are drawn one by one without replacement. 
Number of different ways of selecting $11$ balls out of $25$ balls so that none of drawn balls is red  $$\frac{20}{25}\cdot \frac{19}{24} \cdot \frac{18}{23}\cdot \ldots \frac{11}{16}\cdot\frac{10}{15}=\frac{13}{345}$$
A: I'll break down what you did wrong:
You want to find the probability that all $11$ balls selected are black (not red).
And you're using the complement method which states that the probability (Event A happening) = $1$ - probability (Event A not happening)
e.g. Prob (getting a $2$ after rolling a dice) = $\frac{1}{6}$ or another way to calculate this is to first find the Prob (not getting a $2$ after rolling a dice) = $\frac{5}{6}$ and then subtracting this from $1$. So Prob (getting a $2$ after rolling a dice) = $1- \frac{5}{6} = \frac{1}{6}$ again.
In your case Event A = All $11$ balls are black.
What you have done is first calculated the Prob(A) = Prob (All $11$ balls are black) = $\frac{20}{25} \times \frac{19}{24}\times...\times \frac{10}{15}$ and you should have just left it here. This was the correct answer.
But instead you took this answer and subtracted it from $1$ which now gives you the Prob (all $11$ balls are not black). Hope this clears it out!
