Example - Drawing balls from a box without replacement (sum rule)
Example - Drawing balls from a box without replacement (sum rule)ΒΆ
We now continue our analysis of Example - Drawing balls from a box without replacement. Let us consider the probability of getting a red ball in the second draw without observing in the first draw \(p(B_1|I)\). We have two possibilities for the first draw. We either got a blue ball (B_1 is true) or we got a red ball (R_1 is true). In other words \(B_1\) and \(R_1\) cover all possibilities and are mutually exclusive. We can use the sum rule:
\[\begin{split}
\begin{split}
p(R_2|I) &=& p(R_2|B_1,I)p(B_1|I) + p(R_2|R_1,I)p(R_1|I)\\
&=& \frac{2}{3}\frac{2}{5} + \frac{5}{9}\frac{3}{5}\\
&=& 0.6.
\end{split}
\end{split}\]