1. Meaning of the Multiplication Theorem
The multiplication theorem helps find the probability that two events happen together. It is used for situations involving the word “and”.
Examples:
- Getting a head on a coin and a 4 on a die.
- Drawing a red card and a king.
It calculates the probability of the intersection of two events.
2. General Multiplication Rule
The general form of the multiplication theorem works for any two events A and B.
\( P(A \cap B) = P(A)P(B|A) \)
This means: the chance of A and B both happening equals the chance of A happening first and then B happening after A.
Similarly,
\( P(A \cap B) = P(B)P(A|B) \)
2.1. Example
Suppose:
- A: drawing a red card → 26/52
- B: drawing a king → 4/52
There are 2 red kings.
- \( P(A \cap B) = 2/52 \)
- \( P(A) = 26/52 \)
- \( P(B|A) = 2/26 \)
Using the formula:
\( P(A \cap B) = (26/52)(2/26) = 2/52 \)
3. Multiplication Rule for Independent Events
Two events A and B are independent if the outcome of one does not affect the outcome of the other.
For independent events, the formula becomes simpler:
\( P(A \cap B) = P(A)P(B) \)
This rule is used when the events have no influence on each other.
3.1. Example
Consider:
- A: getting a head on a coin → 1/2
- B: getting a 4 on a die → 1/6
The events do not affect each other.
\( P(A \cap B) = (1/2)(1/6) = 1/12 \)
4. Examples of Dependent and Independent Events
The multiplication theorem is useful for understanding whether events influence each other or not.
4.1. Dependent Events Example
Drawing two cards without replacement:
- A: first card is an ace → 4/52
- B: second card is an ace (after A happens) → 3/51
\( P(A \cap B) = (4/52)(3/51) \)
4.2. Independent Events Example
Rolling two dice:
- A: first die shows a 3 → 1/6
- B: second die shows a 5 → 1/6
\( P(A \cap B) = (1/6)(1/6) = 1/36 \)