# Computing the probability example

I have been doing the following example, and there is something I do not understand. It goes like this:

From an urn with four white balls and six black balls, one ball is drawn repeatedly with replacement until a white ball is obtained. Let X be the number of drawings resulting in a black ball. Compute the probability $$P(X\geq 3)$$.

Now, according to the solutions, the way to solve this is to calculate the following: $$\sum_{k=3}^∞ ((\frac{6}{10})^k\cdot\frac{4}{10})$$

My question: Why cannot I solve it like this: $$P(X\geq 3)=1-(P(X=0)+P(X=1)+P(X=2))$$?

Thank you

• I much prefer your solution! I'd even go a bit further, and say that the answer is the probability of getting $BBB$ as your first three. Makes no difference what happens after that. – lulu Nov 18 '18 at 21:38
• As an editorial note: people often like the Geometric Series method because it's something you can just memorize and work through. Always better to think about the problem and then try to get a simpler solution. – lulu Nov 18 '18 at 21:39
• They are equivalent. It helps to know the general form though. For example, if you had to calculate $P(X \geq 10)$, your method could feel a bit tedious. – Aditya Dua Nov 20 '18 at 7:18