Determine whether events are independent

Independence means that the occurrence of one event does not change the probability of the other, expressed by $P(A\cap B)=P(A)P(B)$ or, when $P(B)>0$ , by $P(A\mid B)=P(A)$ . Determining independence involves comparing these quantities in finite probability models, tables, or tree diagrams and distinguishing independent events from mutually exclusive events, which are generally dependent; treatment is limited to event-based probability rather than more advanced abstract or continuous formulations.

Detailed Explanation: Determine whether events are independent

Two events are independent if knowing that one event occurred does not change the probability of the other. To test this, compare

P(A\cap B)

with

P(A)P(B).

If they are equal, $A$ and $B$ are independent.
If they are not equal, the events are dependent.

Example

Two fair six-sided dice are rolled. Let

$A$ : the first die shows an even number.
$B$ : the sum of the two dice is $7$ .

Determine whether $A$ and $B$ are independent.

Step 1: Find $P(A)$

The first die can show $1,2,3,4,5,$ or $6$ . Three of these are even: $2,4,6$ .

P(A)=\frac{3}{6}=\frac12

Step 2: Find $P(B)$

There are $36$ equally likely outcomes when two dice are rolled. A sum of $7$ occurs in these six ways:

(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)

Therefore,

P(B)=\frac{6}{36}=\frac16

Step 3: Find $P(A\cap B)$

For both events to occur, the first die must be even and the sum must be $7$ .

The possible outcomes are

(2,5),(4,3),(6,1).

Thus,

P(A\cap B)=\frac{3}{36}=\frac{1}{12}

Step 4: Compare with $P(A)P(B)$

P(A)P(B)=\frac12\cdot\frac16=\frac{1}{12}

Since

P(A\cap B)=P(A)P(B),

the events $A$ and $B$ are independent.

Independence does not mean that the events cannot happen together. It means that one event does not change the probability of the other. In contrast, mutually exclusive events cannot happen together, so they are generally dependent.

Learn by doing: Determine whether events are independent

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Determine whether events are independent

Detailed Explanation: Determine whether events are independent

Example

Step 1: Find P(A)P(A)P(A)

Step 2: Find P(B)P(B)P(B)

Step 3: Find P(A∩B)P(A\cap B)P(A∩B)

Step 4: Compare with P(A)P(B)P(A)P(B)P(A)P(B)