Mutually Exclusive Events – The Addition Rule Revision Notes for Leaving Cert Mathematics

Mutually Exclusive Events – The Addition Rule

What are mutually exclusive events?

Mutually exclusive events are outcomes that cannot occur at the same time. When one event happens, the other event cannot happen simultaneously.

Consider drawing a card from the set shown above. Let's look at these two events:

Event A: drawing a red card
Event B: drawing an even number

These events are mutually exclusive because there is no red card with an even number in this set. If you draw a red card, it cannot be an even number, and if you draw an even number, it cannot be red.

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Key point: Mutually exclusive events have no outcomes in common - they cannot happen together. Think of it as events that are completely separate from each other with no overlap.

The addition rule for mutually exclusive events

When events A and B are mutually exclusive, we can find the probability of "A or B" happening using the addition rule:

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The Addition Rule for Mutually Exclusive Events

$P(A \text{ or } B) = P(A) + P(B)$

This is also called the OR rule.

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Worked Example: Card Probabilities

Using the cards from above, let's calculate the probability of drawing a red card or an even number.

Step 1: Set up the calculation $P(\text{red card or even number}) = P(\text{red card}) + P(\text{even number})$

Step 2: Calculate individual probabilities

There are 3 red cards out of 7 total cards: $P(\text{red card}) = \frac{3}{7}$
There are 2 even numbers out of 7 total cards: $P(\text{even number}) = \frac{2}{7}$

Step 3: Apply the addition rule $P(\text{red card or even number}) = \frac{3}{7} + \frac{2}{7} = \frac{5}{7}$

Since the events are mutually exclusive, we can simply add the probabilities.

When events are NOT mutually exclusive

Now consider this different set of cards. What if we want the probability of drawing a red card OR an even number?

Looking at this set:

There are 3 red cards
There are 3 even numbers (4, 6, 8)

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Critical Error to Avoid: Double Counting

The number 4 appears on a red card, so it's counted in both events!

If we simply add the probabilities, we get: $\frac{3}{8} + \frac{3}{8} = \frac{6}{8}$

This is wrong because we've counted the red card with number 4 twice!

The correct answer is $\frac{5}{8}$ because there are only 5 cards that are either red OR even (not 6).

The general addition rule

When two events A and B can occur at the same time, we must use the general formula:

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The General Addition Rule

$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

We subtract $P(A \text{ and } B)$ to avoid counting outcomes that belong to both events twice.

For the cards above: $P(\text{red or even}) = P(\text{red}) + P(\text{even}) - P(\text{red and even})$ $= \frac{3}{8} + \frac{3}{8} - \frac{1}{8} = \frac{5}{8}$

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Worked Example: Multiples

Question: A number is selected at random from the integers 1 to 30 inclusive. Find the probability that the number is: (i) a multiple of 3 (ii) a multiple of 5
(iii) a multiple of 3 or a multiple of 5

Solution:

(i) Multiple of 3: Multiples of 3 from 1 to 30: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 $P(\text{multiple of 3}) = \frac{10}{30} = \frac{1}{3}$

(ii) Multiple of 5: Multiples of 5 from 1 to 30: 5, 10, 15, 20, 25, 30 $P(\text{multiple of 5}) = \frac{6}{30} = \frac{1}{5}$

(iii) Multiple of 3 or multiple of 5: Numbers 15 and 30 are multiples of both 3 and 5, so these events are NOT mutually exclusive.

Using the general formula: $P(\text{multiple of 3 or 5}) = P(\text{multiple of 3}) + P(\text{multiple of 5}) - P(\text{multiple of 3 and 5})$ $= \frac{10}{30} + \frac{6}{30} - \frac{2}{30} = \frac{14}{30} = \frac{7}{15}$

Identifying mutually exclusive events

When working with dice, consider these pairs of outcomes and whether they can occur simultaneously:

First outcome	Second outcome	Mutually exclusive?
Score is 5	Score is 3	✓ Yes - a dice cannot show both 5 and 3 at the same time
Score is 3	Score is even	✓ Yes - 3 is odd, so cannot be even
Score is even	Score is greater than 4	✗ No - 6 is both even and greater than 4
Score is prime	Score is even	✗ No - 2 is both prime and even
Score is multiple of 5	Score is multiple of 3	✓ Yes - no number from 1-6 is both

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Testing for Mutual Exclusivity

Use the word "simultaneously" to test mutual exclusivity:

Ask: "Can these two events happen simultaneously?"
If no, they're mutually exclusive
If yes, use the general addition formula

Common exam traps and tips

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Trap 1: Assuming events are mutually exclusive when they're not

Always check if outcomes can overlap before using the simple addition rule. Look carefully for numbers, cards, or outcomes that satisfy both conditions.

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Trap 2: Double counting in "or" questions

When events overlap, you must subtract the intersection. Count carefully to avoid including outcomes twice - this is the most common mistake in probability questions.

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Exam Success Tips

Always check whether events can occur simultaneously before choosing which formula to use
List out the outcomes when in doubt - it helps visualise overlaps
The addition rule is also known as the OR rule - it calculates the probability of at least one of the events happening

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Key Points to Remember:

Mutually exclusive events cannot happen at the same time - they have no outcomes in common
For mutually exclusive events: $P(A \text{ or } B) = P(A) + P(B)$
For events that can overlap: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$
Always check whether events can occur simultaneously before choosing which formula to use
The addition rule is also known as the OR rule - it calculates the probability of at least one of the events happening

Mutually Exclusive Events – The Addition Rule (Leaving Cert Mathematics): Revision Notes