Conditional Probability

What is conditional probability?

$P(A|B)$ is the probability of $A$ given that $B$ has already occurred. It restricts the sample space to the outcomes where $B$ happened.

$P(A|B) = \frac{P(A \cap B)}{P(B)}$

Step-by-step: Drawing without replacement

Setup: Bag has 4 red + 3 blue = 7 balls.

Step 1: First draw is red. Now the bag has 3 red + 3 blue = 6 balls remaining.

Step 2: $P(\text{2nd red} | \text{1st red}) = \frac{3}{6} = \frac{1}{2} = 50\%$

Note: Without the "given" information, $P(\text{any single ball is red}) = 4/7 \approx 57\%$. After removing one red ball, the probability drops to $3/6 = 50\%$.

With replacement vs without

• With replacement: $P(\text{2nd red}) = 4/7$ always (bag is reset).
• Without replacement: $P(\text{2nd red} | \text{1st red}) = 3/6 = 1/2$ (bag has changed).

Independence

Events are independent if knowing one happened doesn't change the probability of the other: $P(A|B) = P(A)$. Drawing with replacement → independent. Drawing without replacement → dependent.

Multiplication rule

$P(A \cap B) = P(A) \cdot P(B|A)$

$P(\text{1st red AND 2nd red}) = \frac{4}{7} \times \frac{3}{6} = \frac{12}{42} = \frac{2}{7} \approx 28.6\%$

Try it in the visualization

The bag shows colored balls. Drawing one ball updates the bag visually. The probability of the next draw changes based on what was removed. Toggle between with/without replacement to see the difference.