Multiplication Rule (Independent)

Statement

The multiplication rule for independent events allows us to calculate the probability of two independent events happening together. Two events are said to be independent if the occurrence of one has no effect on the probability of the other. For such events:

\[ P(A \cap B) = P(A) \times P(B) \]

This means the probability that both \(A\) and \(B\) occur is simply the product of their individual probabilities.

Why it’s true

• If events are independent, knowing that one occurred does not change the likelihood of the other. For example, flipping a fair coin and rolling a fair dice are independent actions.
• The rule reflects the basic principle that for independent events, the combined chance is the product of their individual chances.

Recipe (how to use it)

1. Check that the events are independent (e.g., one is a coin toss, the other is a dice roll).
2. Write down the probability of each event separately.
3. Multiply the probabilities together.
4. Simplify the fraction or decimal if possible.

Spotting it

Look for questions that mention experiments with no connection to each other, such as tossing two coins, drawing two balls with replacement, or independent trials in probability experiments.

Common pairings

• Coin tosses and dice rolls.
• Repeated independent trials (e.g., flipping a coin 3 times).
• Probabilities with replacement (e.g., drawing a card, putting it back, then drawing again).

Mini examples

1. Given: Probability of rolling a 6 on a dice is \(1/6\). Probability of flipping heads is \(1/2\). Find: both happening. Answer: \(1/6 \times 1/2 = 1/12\).
2. Given: Probability of winning a game is \(0.3\). Probability of raining tomorrow is \(0.4\). Find: both occurring. Answer: \(0.3 \times 0.4 = 0.12\).

Pitfalls

• Using the multiplication rule for dependent events (e.g., drawing cards without replacement). In such cases, probabilities change after each event.
• Forgetting to check independence before multiplying.
• Adding probabilities instead of multiplying when “and” is required.

Exam strategy

• Look carefully for the word “independent”.
• If probabilities are fractions, keep them exact rather than converting into decimals unless told.
• Check that your final answer is between 0 and 1 — probabilities cannot exceed these bounds.

Summary

The multiplication rule for independent events is a cornerstone of probability. It allows us to calculate the likelihood of two unrelated events occurring at the same time by multiplying their individual probabilities. The key is independence: if events affect each other, this rule no longer applies.