Question

I have 3 cups with 3 marbles each; one black and two white. If I draw a marble from each cup, what is the probability that I will draw AT LEAST ONE white marble? With one cup it's 66%. If I have 100 cups, the probability that I will draw at least 1 white marble is nearly 190%. How do you calculate the actual probability of drawing at least one white marble, given N cups?

Answer 1

Let $B$ be the event "draw a black marble from a cup". Then

$$1 \ cup:P(B)={1 \over 1+2}={1 \over 3}$$

$$2 \ cups:P(BB)={1 \over 3}\cdot{1 \over 3}=({1 \over 3})^2$$

$$3 \ cups:P(BBB)=({1 \over 3})^3$$

$$...$$

$$N \ cups: P(B\ \text{n times})=({1 \over 3})^N$$

$$P(\text{at least 1 white marble if I have N cups)}=$$

$$=1-P(B\ \text{n times})=1-({1 \over 3})^N$$

$$100\ cups:P(\text{at least 1 white marble if I have 100 cups)}=$$

$$=1-({1 \over 3})^{100}\approx$$

$$\approx0.99999999999999999999999999999999999999999999999806$$