1.2 Joint Probability

• Definition: Probability of two specific events happening together.
• Example (Balls in a Bag):
• A bag has 4 Red Balls (R) and 6 Blue Balls (B). Two balls are drawn.
• Joint Probability of drawing Red first AND Blue second (without replacement):

$$ P(R \cap B) = P(R) \times P(B|R) = \frac{4}{10} \times \frac{6}{9} = \frac{24}{90} = 0.266 $$

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2. Random Variables

A variable whose value depends on the outcome of a random experiment.

2.1 Types of Random Variables

1. Discrete Random Variable:

• Takes specific, countable values.
• Example: You flip a coin 3 times. Let $X$ = Number of Heads.
• Possible values for $X$: ${0, 1, 2, 3}$. (You cannot get 2.5 heads).

2. Continuous Random Variable:

• Takes any value within a range (infinite possibilities).
• Example: Let $Y$ = Height of a student in a class.
• Possible values: Could be 150.1 cm, 150.12 cm, 160.5 cm, etc. It is measured, not counted.