Mutually Exclusive Events
Some events cannot happen at the same time. These are called mutually exclusive events. Understanding this idea allows probabilities to be combined correctly.
What Mutually Exclusive Means
Two events are mutually exclusive if they cannot both occur in the same experiment.
If one event happens, the other cannot happen.
For example:
• rolling a 2 and rolling a 5 on a single die
• getting heads and tails on one coin toss
These events have no outcomes in common.
Events that can happen together are not mutually exclusive
Probability of A or B
When events A and B are mutually exclusive, the probability that A or B occurs is found by adding their probabilities.
This is written as:
\( P(A\ or\ B) = P(A) + P(B) \)
This rule works because there is no overlap between the events.
Nothing is counted twice.
Why the Rule Works
Because A and B cannot occur together:
• all outcomes that make A happen are separate from B
• all favourable outcomes are distinct
So the total probability is simply the sum of the two individual probabilities.
If the probabilities were added for events that were not mutually exclusive, the overlap would be counted twice.
That would give an incorrect result.
Using the Rule Correctly
This rule can only be used when:
• events are mutually exclusive
• probabilities are written on the same scale
• all probabilities are valid values between 0 and 1
If events are not mutually exclusive, a different rule must be used.
Always check whether events can occur together
Common Errors to Avoid
Common mistakes include:
• adding probabilities for events that overlap
• assuming events are mutually exclusive without checking
• forgetting that the rule only applies to A or B, not A and B
Careful reading of the question helps avoid these errors.
Key Points to Remember
Mutually exclusive events cannot occur at the same time.
They have no outcomes in common.
For mutually exclusive events A and B:
\( P(A\ or\ B) = P(A) + P(B) \)
This rule only applies when there is no overlap.
Always check the conditions before adding probabilities.
Understanding mutually exclusive events ensures probabilities are combined correctly and avoids common mistakes when analysing chance situations.