Q.An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted, and it is returned to the urn. Moreover, 2 additional balls of the colour drawn are put into the urn and then a ball is drawn at random. The probability that the second ball drawn is red is

Theorem 12.10 (Total Probability of an Event). Suppose $A_1,A_2,\ldots,A_n$ are mutually exclusive AND exhaustive events (a partition of the sample space $S$), and $B$ is any event in $S$. Then

Why it works. Since $A_1,\ldots,A_n$ partition $S$, the event $B$ splits into the mutually exclusive pieces $(A_1\cap B),(A_2\cap B),\ldots,(A_n\cap B)$, whose union is $B$ itself. Adding their probabilities (axiom $[P_2]$) gives $P(B)=\sum_i P(A_i\cap B)$, and the Multiplication Theorem rewrites each term as $P(A_i)P(B/A_i)$. …