Let’s consider a situation where we have two bags (Bag A and Bag B) with the following distribution of colored balls:

• Bag A: 54 Blue balls, 6 Red balls
• Bag B: 20 Blue balls, 20 Red balls

In total, there are 54 + 6 + 20 + 20 = 100 balls. We will use this setup to illustrate several probability concepts: joint probability, marginal probability, law of total probability, and conditional probability.

To simplify the analysis, we make a naive assumption (the classic probability model), where each sample has an equal likelihood of being selected. Under this assumption, the probability of selecting Bag A is $60/100 = 0.60$, and the probability of selecting Bag B is $40/100 = 0.40$.

While this assumption aids in understanding the concepts, it may not always hold true in real-world scenarios.

Joint Probability

Joint probability refers to the probability of two events occurring simultaneously. We denote it as $P(A,B)$.

1. Setting Up the Events
   • Let "Bag A" be event $A$; and let "Blue" be event $B$.
   • Then $P(\text{Bag A, Blue})$ or $P(A, B)$ is the probability of picking a ball that is both from Bag A and Blue in one random draw.

Marginal Probability

Marginal probability is the probability of one event, ignoring other distinctions. For example, $P(\text{Blue})$ is the probability of drawing a blue ball from either bag, with no regard to which bag it came from.

Often, we find marginal probability by summing relevant joint probabilities.

Counting Perspective (Frequency)

• Raw count of blue balls: 54 in Bag A + 20 in Bag B = 74 total blue balls.
• Total of 100 balls, so $\frac{74}{100} = 0.74$.

The Law of Total Probability: it states that if an event can occur via several disjoint “routes,” you sum over those routes:

$$ P(\text{Blue})= P(\text{Bag A, Blue}) + P(\text{Bag B, Blue}). $$

1. Calculating $P(\text{Bag A, Blue})$ and $P(\text{Bag B, Blue})$

   • We already have $P(\text{Bag A, Blue}) = 0.54.$

   • For Bag B, we have

     $$ P(\text{Bag B}) = \tfrac{40}{100} = 0.40,\quad P(\text{Blue}\mid \text{Bag B}) = \tfrac{20}{40} = 0.50. $$

   • Thus,

     $$ P(\text{Bag B, Blue}) = 0.40 \times 0.50 = 0.20. $$

2. Combining via the Law of Total Probability

   $$ P(\text{Blue}) = 0.54 + 0.20 = 0.74. $$

   • So there is a 74% chance of drawing a blue ball overall.

Conditional Probability

Conditional probability is the probability of an event $A$ occurring given that another event $B$ has already occurred. It's denoted as $P(A|B)$, which reads as "the probability of A given B."

$$ P(A \mid B) = \frac{P(A, B)}{P(B)}. $$