Introduction to Combined Events

In your study of probability so far, you have likely looked at single events, like rolling a die once or picking one card. But what happens when we do more than one thing? For example, what is the probability of rolling a \(6\) and flipping a Head? These are called combined events.

In this chapter, we will explore the two main types of combined events: independent and dependent. Understanding the difference between these is the "secret key" to getting probability questions right every time!

1. Independent Events

Events are independent if the outcome of the first event has no effect whatsoever on the outcome of the second event. The events simply don't care about each other!

Example: If you roll a die and flip a coin, the result of the die doesn't change the chances of the coin landing on Heads. They are independent.

The "AND" Rule for Independent Events

When you want to find the probability of event \(A\) and event \(B\) happening, you multiply their individual probabilities.

The Formula: \(P(A \text{ and } B) = P(A) \times P(B)\)

Step-by-Step Example:

Imagine you roll a fair six-sided die and flip a fair coin. What is the probability of getting a \(4\) on the die and a Tail on the coin?

1. Find the probability of the first event: \(P(4) = \frac{1}{6}\)
2. Find the probability of the second event: \(P(\text{Tail}) = \frac{1}{2}\)
3. Multiply them together: \(\frac{1}{6} \times \frac{1}{2} = \frac{1}{12}\)

Did you know? The word "independent" usually applies to situations involving replacement. If you pick a marble from a bag, look at it, and put it back before picking again, the second pick is independent of the first!

2. Dependent Events

Events are dependent if the outcome of the first event does change the probability of the second event. This usually happens in "without replacement" problems.

Example: Imagine a bag contains \(3\) red sweets and \(7\) blue sweets. If you eat a red sweet, there are now fewer sweets in the bag and fewer red ones left. The "world" has changed for your second pick!

Calculating Dependent Probabilities

To solve these, you still multiply, but you must update the second probability to reflect what happened in the first event.

Higher Tier Note: On your formula sheet, this is written as \(P(A \text{ and } B) = P(A \text{ given } B) \times P(B)\). This just means you multiply the probability of the first event by the probability of the second event after the first has happened.

Step-by-Step Example:

A box contains \(5\) green pens and \(3\) black pens. You pick two pens at random without replacement. What is the probability that both pens are green?

1. Probability the first pen is green: \(P(\text{Green 1}) = \frac{5}{8}\) (there are \(5\) green out of \(8\) total).
2. Stop and think: If the first was green, there are now only \(4\) green pens left and only \(7\) pens in total.
3. Probability the second pen is green: \(P(\text{Green 2}) = \frac{4}{7}\).
4. Multiply them: \(\frac{5}{8} \times \frac{4}{7} = \frac{20}{56}\). (You can simplify this to \(\frac{5}{14}\)).

Key Takeaway:

Always check if a question says "with replacement" (Independent) or "without replacement" (Dependent).

3. "AND" vs "OR"

Students often get confused about when to multiply and when to add. Here is a simple rule of thumb to remember:

  • AND means Multiply: If you want event A and event B to happen, multiply the probabilities.
  • OR means Add: If you are happy with either event A or event B happening, add the probabilities.

Formula Sheet Alert: For all students, the formula sheet provides: \(P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\). This is used when events are not mutually exclusive (meaning they could happen at the same time).

4. Common Pitfalls and Tips

Don't worry if this feels like a lot to track! Here are some common mistakes to watch out for:

  • Forgetting to change the denominator: In "without replacement" (dependent) questions, the bottom number of your fraction must decrease for the second event.
  • Assuming independence: Read the question carefully. If it doesn't say "with replacement," look for clues like "eats a sweet," "keeps the card," or "takes two items at the same time" (which is the same as taking one and not replacing it).
  • Decimal vs Fraction: You can use either! However, fractions are often easier for "without replacement" problems because you can see the numbers decreasing clearly.

Quick Review Box

Independent: Outcomes do not affect each other. (Example: Coins, Dice, With Replacement).
Dependent: The first outcome changes the total for the second. (Example: Choosing socks from a drawer, Without Replacement).
The Golden Rule: For combined events happening one after another, multiply the probabilities of each stage.

Note: For help on how to draw these out visually, see the chapter on Sample spaces and Tree diagrams, where we use these multiplication rules to fill out the branches!