9.2 Conditional Probability and Independence

Section 9.1 measured events inside a complete sample space. New information can rule out some outcomes. When that happens, the denominator must change because the set of outcomes still under consideration has changed.

Suppose a card is chosen uniformly from

Before receiving any information, the event $A=$ “the card is greater than $5$” is $\{6,7,8\}$, so $P(A)=3/8$.

Now someone tells us that the selected card is even. Let

The possible cards are no longer all eight members of $Omega$; only the four members of $B$ remain possible. Among them, $6$ and $8$ also belong to $A$. The revised probability is therefore $2/4=1/2$.

Conditional probability restricts the sample space

The probability of $A$ given that $B$ has occurred is called a conditional probability. It is written

The vertical bar is read “given.” It is not division. The event on the right of the bar is the information we are assuming.

Inside the restricted space $B$, the successful outcomes are exactly $A\cap B$. This gives the definition

The condition $P(B)>0$ is necessary because division by zero is undefined. In a finite equally likely sample space, the probability ratio becomes a count ratio:

For the card example,

The order around the bar matters

$P(A\mid B)$ and $P(B\mid A)$ usually answer different questions.

For the same cards,

because after learning that the card is greater than $5$, the restricted space is $A=\{6,7,8\}$. The numerator is the same intersection, but the denominator is different.

Reading conditional probability from a table

A two-way table organizes two yes-or-no events. Imagine $100$ students classified by whether they use Python ($A$) and SQL ($B$):

To find $P(A\mid B)$, first restrict attention to the $B$ column. Its total, $40$, becomes the denominator. The $A\cap B$ cell, $30$, becomes the numerator:

To find $P(B\mid A)$, use the $A$ row instead:

A dependable rule is: the condition chooses the denominator.

Operate a filter lens over a population grid. Define two events by changing thresholds, then place either event behind the conditioning bar. Outcomes outside the condition fade away, leaving the new denominator and intersection visible.

The multiplication rule reverses the definition

Starting from

multiply both sides by $P(B)$:

The same intersection can be reached in the opposite order:

This is the multiplication rule. It describes a two-stage route: first reach the condition, then reach the second event inside that restricted branch.

For example, a box contains $3$ red and $2$ blue tokens. Draw two tokens uniformly without replacement, meaning the first token is not returned before the second draw. Let $R_1$ mean “the first token is red” and $R_2$ mean “the second token is red.” Then

After a red token is drawn, only $2$ red tokens remain among $4$ total tokens, so

Thus

A probability tree is a branching diagram for this multiplication. Each branch records a conditional probability, and the probability of a complete root-to-end route is the product of its branch labels. This is only a diagramming convention; the formal rule is the multiplication equation above.

Independence means information does not change probability

Events $A$ and $B$ are independent when learning that one occurred does not change the probability of the other. When $P(B)>0$, this idea can be written

Substituting the conditional-probability definition gives the symmetric test

This product equation is usually the safest definition because it also works when one event has probability $0$.

Example: two separate coin tosses

Let $A$ be “the first toss is heads” and $B$ be “the second toss is heads.” In

we have

Because

the events are independent.

Example: drawing without replacement

In the token box, $P(R_2)=3/5$ before any draw because each token is equally likely to occupy the second position. But

The first red draw changes the composition of the box, so $R_1$ and $R_2$ are dependent.

Mutually exclusive is not the same as independent

Events are mutually exclusive if they cannot occur together:

For one die roll, “roll a $2$” and “roll a $5$” are mutually exclusive. Observing a $2$ makes a $5$ impossible on that same roll, so the events strongly affect each other.

If mutually exclusive events both have positive probability, then

Therefore they are not independent. Independence allows overlap at exactly the rate predicted by the product of the separate probabilities; mutual exclusion forbids overlap entirely.

Redistribute a fixed population among the four cells of a two-way table. A live balance compares the observed overlap with the overlap predicted by independence, while a cross-product seal gives an exact integer test with no rounding.

Section checklist

• The event after the bar is the restricted sample space.
• $P(A\mid B)$ and $P(B\mid A)$ usually have different denominators.
• The multiplication rule reconstructs an intersection from a route.
• Independence means $P(A\cap B)=P(A)P(B)$.
• Sampling without replacement usually creates dependence.
• Mutually exclusive positive-probability events are not independent.

Conditional probability tells us how evidence changes a probability. In the next section, Bayes' theorem will reverse the direction of a condition: it will turn a known probability of evidence under a cause into the probability of that cause after seeing the evidence.