Probability of selecting a red ball first An urn contains 3 red and 7 black balls. Players A
and B withdraw balls from the urn consecutively
until a red ball is selected. Find the probability that
A selects the red ball. (A draws the first ball, then
B, and so on. There is no replacement of the balls
drawn.)
How do I calculate this probability?
I tried using the total probability rule without success.
I used the $P(A)=\frac{3}{10}+P(A_2\mid B_1)$ and so on, where $B_i$=Player B doesn't get a red ball.
The answer should be $0.0888$

Here's what I was thinking.
$$\color{RED}R+BB\color{RED}R+BBBB\color{RED}R+BBBBBB\color{RED}R$$
$$P(A)=\frac{3}{10}+\frac{7}{10}\frac{6}{9}\frac{3}{8}+\frac{7}{10}\frac{6}{9}\frac{5}{8}\frac{4}{7}\frac{3}{6}+\frac{7}{10}\frac{6}{9}\frac{5}{8}\frac{4}{7}\frac{3}{6}\frac{2}{5}\frac{3}{4}=\frac{7}{12}$$
