Some events do not affect each other. These are called independent events. Understanding this allows probabilities to be combined correctly when events happen together.

 

 

What Independent Means

Two events are independent if the occurrence of one event does not change the probability of the other event occurring.

 

The outcome of one event has no influence on the outcome of the other.

 

For example:
• tossing a coin and rolling a die
• tossing two separate coins

 

The result of one trial does not affect the other

 

 

Probability of A and B

When events A and B are independent, the probability that A and B both occur is found by multiplying their probabilities.

 

This is written as:

\( P(A\ and\ B) = P(A) \times P(B) \)

 

This rule applies because the chance of both events happening depends on both probabilities occurring together.

 

 

Why the Rule Works

Because the events are independent:
• the probability of A stays the same whether or not B happens
• the probability of B stays the same whether or not A happens

 

The combined probability is therefore the product of the two individual probabilities.

 

Multiplication reflects the idea that both events must occur.

 

 

Using the Rule Correctly

This rule can only be used when:
• events are independent
• probabilities are written on the same scale
• probabilities lie between zero and one

 

If events are not independent, this rule cannot be used.

 

Always check whether events affect each other

 

 

Common Errors to Avoid

Common mistakes include:
• multiplying probabilities for events that are not independent
• confusing independent events with mutually exclusive events
• adding probabilities instead of multiplying

 

Mutually exclusive events use a different rule.

 

 

Key Points to Remember

Independent events do not affect each other.
For independent events A and B, the probability that both occur is found by multiplying.
The rule is written as \( P(A\ and\ B) = P(A) \times P(B) \).
This rule only applies when events are independent.
Always check the relationship between events before combining probabilities.

 

Understanding independent events ensures probabilities are calculated correctly when analysing situations involving repeated or unrelated trials.