Suppose, there are $5$ Red balls, $2$ Green Balls and $3$ Yellow Balls in a bag. What is the probability that the second ball taken out is a Red ball (the color of the first ball taken out can be any of the three)?
I have made the outcome tree and came up with the answer as $\frac12$.
But I don't know if it is correct.
Best Answer
If the first ball is red (which happens with probability $\frac{5}{10}=\frac{1}{2}$) we are left with $9$ balls of which $4$ red, so then the probability is $\frac{4}{9}$ for the second one to be red.
If the first is non-red (which also happens with probability $\frac{5}{10}=\frac{1}{2}$) we're left with $9$ balls of which $5$ are red, so then the probability is $\frac{5}{9}$ for the second one to be red.
So the "bare" or "full" probability of the second one being red is computed by conditioning on these two first outcomes and equals
$$\frac{1}{2}\times \frac{4}{9} + \frac{1}{2}\times \frac{5}{9}= \frac{9}{18}=\frac{1}{2}$$
so your answer is correct.