When we start learning about advanced probability, especially the rules for adding and multiplying probabilities, there are some misunderstandings that can confuse students. Here’s a simple breakdown of what I’ve noticed:

1. Confusion About the Addition Rule

• Exclusive vs. Overlapping Events: A common mistake is thinking the addition rule works the same for all types of events.

  For exclusive events (where they can’t happen at the same time, like flipping a coin and getting heads or tails), we just add the probabilities:

  • If we say $P(A \text{ or } B)$, it means $P(A) + P(B)$.

  But when events overlap (they can happen at the same time), we need to be careful. We have to subtract the chance of both happening at once so we don’t count it twice:

  • Here, $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$.

• Thinking Events Are Independent: Some students think events are independent without checking. Remember, two events are independent if one doesn’t affect the other. If they are dependent, simply adding the probabilities doesn’t work!

2. Confusion About the Multiplication Rule

• Using the Multiplication Rule Right: The multiplication rule can also be tricky.

  For independent events, the chance of both happening is just the product of their probabilities:

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

  But if the events are dependent, we have to change how we calculate it:

  • Then it becomes $P(A \text{ and } B) = P(A) \times P(B|A)$.

• Assuming All Events Are Independent: A lot of students quickly use the multiplication rule for everything, which can lead to wrong answers. Just because two events seem like they could be independent doesn’t mean they actually are.

3. Ignoring Complementary Events

• Another misunderstanding is ignoring complementary events in probability.

  It’s important to know that the chance of something not happening is $1 - P(A)$. This is really useful when we’re figuring out the probabilities of different events.

These misunderstandings can make learning and using probability harder. It’s important to look closely and understand the situation of the events we are working with.