When you start learning about probability, two important types of events are independent and dependent events. Knowing the difference between these events is really important, especially for your Year 12 AS-Level studies.

Independent Events

Let’s begin with independent events.

These are events that don’t affect each other.

In simple words, if one event happens, it doesn’t change what happens with the other event.

For example, think about flipping a coin and rolling a die at the same time:

• Coin Flip: You can get heads or tails.
• Die Roll: You can roll a number from 1 to 6.

No matter what side the coin shows, the die still has the same chances of showing any number.

To calculate the chances of both happening together, you can use this multiplication rule:

$P(A \text{ and } B) = P(A) \times P(B)$

Here’s a simple example with numbers:

If you flip a coin (which has a 50% chance of being heads) and roll a die (which has a 1 in 6 chance for each number), the chance of getting heads and a 4 is:

$P(\text{Heads and 4}) = P(\text{Heads}) \times P(4) = 0.5 \times \frac{1}{6} = \frac{1}{12}$

Dependent Events

Now, let’s talk about dependent events.

In this case, the result of one event does change the result of another.

A great example is drawing cards from a deck. If you pick one card and leave it out, the next draw will be affected by what you drew first.

Imagine you draw an Ace from a regular 52-card deck.

If you don’t put the Ace back, now there are only 51 cards left for your next draw.

If event A is drawing an Ace and event B is drawing a King, the chances change for the second event because of the first:

$P(B | A) = \frac{\text{Number of Kings}}{\text{Total cards left}} = \frac{4}{51}$

In this equation, the pipe symbol ($|$) shows that we are looking at the chance of event B happening after event A has already happened.

So, if you already drew an Ace, the chance of drawing a King now changes.

Summary of Key Differences

Here’s a quick summary of the differences:

• Independent Events:

  • Definition: Events that do not affect each other.
  • Example: Coin flip and die roll.
  • Rule: $P(A \text{ and } B) = P(A) \times P(B)$.

• Dependent Events:

  • Definition: Events where one event affects the outcome of another.
  • Example: Drawing cards without replacement.
  • Rule: $P(B | A) = \frac{P(A \text{ and } B)}{P(A)}$.

Conclusion

Knowing whether events are independent or dependent is very important for calculating probabilities correctly.

It’s like having a cheat sheet for different sections in a video game!

Every type of event needs a different way of using the probability rules.

Getting this right will help you solve problems better and make you feel more confident as you work through your Year 12 maths studies.

Happy studying!