1. Meaning of Independent Events
Two events are called independent if the occurrence of one does not affect the occurrence of the other. In simple terms, knowing that one event has happened gives no information about the other.
Examples of independent situations:
- Tossing a coin and rolling a die.
- Drawing a card from a deck, replacing it, and then drawing again.
- Choosing a person from one group and a person from another group.
2. Definition of Independent Events
Events A and B are independent if:
\( P(A|B) = P(A) \)
and
\( P(B|A) = P(B) \)
This means that knowing B happened does not change the chance of A, and vice versa.
3. Multiplication Rule for Independent Events
For independent events, the multiplication rule becomes:
\( P(A \cap B) = P(A)P(B) \)
This formula gives the probability that both events happen together.
3.1. Example
Consider the events:
- A: getting a head on a coin → 1/2
- B: getting a 3 on a die → 1/6
Since the events do not affect each other:
\( P(A \cap B) = (1/2)(1/6) = 1/12 \)
4. Examples of Independent Events
Here are some simple examples to understand independence better:
4.1. Coin and Dice Example
Tossing a coin and rolling a die are independent because the coin result does not influence the die number.
4.2. Cards With Replacement
Drawing a card, replacing it back into the deck, and then drawing again creates independent events because the deck returns to its original state.
4.3. Two Separate Bags
Picking a ball from Bag A and then picking a ball from Bag B forms independent events, since the two bags do not interact.
5. When Events Are Not Independent
Events are not independent (they are dependent) when one affects the other. For example:
- Drawing two cards without replacement — the first draw affects the second.
- Choosing students from the same group — the first choice changes who is left.
In such cases, independence does not hold and conditional probability must be used instead.