Use of Conditional Probability Example
Bag 2 has two toffees.
Bag 3 has one chocolate and one toffee.
From a randomly chosen bag, you choose a sweet at random, which turm out to be a chocolate. What is the probability that the other sweet in the bag is a chocolate?
First find the probability that you pick a chocolate.
\[\begin{equation} \begin{aligned} P(Chocolate) &= P(You \: pick \: bag \: 1 \: then : a \: chocolate) + P(You \: pick \: bag \: 2 \: then : a \: chocolate) \\ &+ P(You \: pick \: bag \: 3 \: then : a \: chocolate) \\ &= \frac{1}{3} \times 1 + \frac{1}{3} \times 0 + \frac{1}{3} \times \frac{1}{2} \\ &= \frac{1}{2} \end{aligned} \end{equation}\]
Now
\[P(Other \: Sweet \: in \: Bag \: is \: Chocolate)=\frac{P(You \: Chose \: a \: Chocolate \: From \: Bag \: 1)}{P(You \: Chose \: a \: Chocolate)}=\frac{1/3}{1/2}= \frac{2}{3}\]
