Understanding Probability: Common Mistakes to Avoid

When you start learning about probability in Year 1 Gymnasium, it’s really important to get the basics right. However, students often make some common mistakes, especially with two main rules: the addition rule and the multiplication rule. Let’s look at these mistakes together!

1. Getting the Addition Rule Wrong

The addition rule helps us find the chance of either one event or another happening. A common error is using this rule incorrectly when the events can happen at the same time.

Example:

Think about rolling a die (a cube with numbers 1 to 6).

• Let’s say event A is rolling a 2.
• Event B is rolling an even number (which means 2, 4, or 6).

If you want to find the chance of rolling either a 2 or an even number, you might wrongly just add the chances of A and B together.

• The chance of A (rolling a 2) is $\frac{1}{6}$.
• The chance of B (rolling an even number) is $\frac{3}{6}$.

If you add these, you get:

$\frac{1}{6} + \frac{3}{6} = \frac{4}{6}$

But that’s not right! Since rolling a 2 is part of rolling an even number, you need to subtract the chance of A to avoid counting it twice:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

So it should look like this:

$P(A \cup B) = \frac{1}{6} + \frac{3}{6} - \frac{1}{6} = \frac{3}{6}$

2. Not Checking If Events Are Independent

The multiplication rule is used to find the chance of two independent events happening together. A common mistake is assuming that the events don’t affect each other without checking first.

Example:

Think about two events:

• Event A: the sun is shining.
• Event B: you get a good grade on a math test.

Some students might quickly use the multiplication rule to find the chance of both happening, without considering if one could influence the other.

For independent events, you can multiply their chances:

$P(A \cap B) = P(A) \cdot P(B)$

But if it’s a cloudy day and you feel low because of the weather, A and B are not independent anymore.

3. Forgetting the Sample Space

Sometimes, students forget to clearly define their sample space. The sample space includes all possible outcomes that the probabilities relate to.

Example:

If you’re drawing a card from a deck, the sample space is all 52 cards. If you want to find the chance of drawing an Ace, remember:

$P(\text{Ace}) = \frac{4}{52}$

Not recognizing all possible outcomes can lead to mistakes in calculating probabilities.

4. Confusing “At Least One” With Exact Outcomes

When dealing with situations that say “at least one,” students often mess up the calculations.

Example:

To find the chance of rolling at least one 2 in two rolls of the die, some think it’s just the chance of rolling a 2 in one roll two times. They might forget there are other combinations.

Instead, it’s often easier to calculate the opposite: find the chance of not rolling a 2 at all, then subtract that from 1. The chance of not rolling a 2 with one die is $\frac{5}{6}$.

So, it looks like this:

$P(\text{at least one 2}) = 1 - P(\text{not rolling a 2})^2 = 1 - \left( \frac{5}{6} \right)^2$

This equals:

$1 - \frac{25}{36} = \frac{11}{36}$

Conclusion

Getting a good grip on these rules in probability helps build a strong base for your math education. By avoiding these common mistakes—like misusing the addition rule, not checking if events are independent, ignoring the sample space, and mixing up “at least one” scenarios—you can tackle probability problems more confidently and accurately. Keep practicing, and don’t hesitate to ask for help if you’re unsure!