Mutually Exclusive Events
Section: Probability | Syllabus: Cambridge Lower Secondary Checkpoint Mathematics (0862)
What are Mutually Exclusive Events?
Mutually exclusive events cannot happen at the same time - if one occurs, the other cannot.
- Rolling a 3 OR a 5 on a die: these can't both happen on the same roll - mutually exclusive.
- Drawing a King OR a Heart from a deck: these CAN both happen (the King of Hearts) - NOT mutually exclusive.
The Addition Rule for Mutually Exclusive Events
For mutually exclusive events A and B: P(A or B) = P(A) + P(B).
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Question: A bag contains 12 balls: 5 red, 4 blue, 3 green. Find P(red or blue).
- Step 1: P(red) = 5/12, P(blue) = 4/12
- Step 2: P(red or blue) = 5/12 + 4/12 = 9/12
- Answer: 3/4 (simplified)
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Question: A spinner has 8 equal sections numbered 1-8. Find the probability of spinning a 2 or a 7.
- Step 1: P(2) = 1/8, P(7) = 1/8
- Step 2: P(2 or 7) = 1/8 + 1/8 = 2/8
- Answer: 1/4 (simplified)
Probabilities from a List with Repeated Values
When outcomes in a list are NOT all different, count how many times each one appears before applying the addition rule - the probability of a repeated outcome is its own count divided by the total.
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Question: The letters of the word BANANA are written on 6 separate cards, one letter per card. A card is chosen at random. Find P(A or N).
- Step 1: BANANA has 3 A's, 2 N's, and 1 B, out of 6 letters: P(A) = 3/6, P(N) = 2/6
- Step 2: A card can't show both letters at once, so A and N are mutually exclusive: P(A or N) = 3/6 + 2/6
- Answer: 5/6
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Question: Using the same 6 cards, find P(not A).
- Step 1: P(A) = 3/6 = 1/2
- Step 2: P(not A) = 1 − 1/2
- Answer: 1/2
All Possible Outcomes Sum to 1
The complete set of mutually exclusive outcomes for an event covers everything that could happen, so their probabilities always add up to exactly 1. This lets you find a missing probability by subtracting the known ones from 1.
All mutually exclusive outcomes fill the entire probability bar - they sum to 1
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Question: A weighted die has P(1)=0.1, P(2)=0.15, P(3)=0.2, P(4)=0.25, P(5)=0.1. Find P(6).
- Step 1: Sum of known probabilities: 0.1+0.15+0.2+0.25+0.1 = 0.8
- Step 2: P(6) = 1 − 0.8
- Answer: P(6) = 0.2
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Question: A bag has only red, blue, and green balls. P(red)=0.4, P(blue)=0.35. Find P(green).
- Step 1: P(green) = 1 − 0.4 − 0.35
- Answer: P(green) = 0.25
Combining Multiple Mutually Exclusive Events
The addition rule extends to any number of mutually exclusive events: P(A or B or C) = P(A) + P(B) + P(C).
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Question: A spinner has 10 equal sections: 3 red, 2 blue, 4 green, 1 yellow. Find P(red or blue or yellow).
- Step 1: P(red) = 3/10, P(blue) = 2/10, P(yellow) = 1/10
- Step 2: P(red or blue or yellow) = 3/10 + 2/10 + 1/10 = 6/10
- Answer: 3/5 (simplified)
Real-World Applications
Mutually exclusive probabilities appear in many contexts:
- Weather forecasting: probabilities of sunny/cloudy/rainy sum to 1.
- Games and gambling: dice, cards, and spinners.
- Quality control: probability of a product having different specific defects.
- Elections: probabilities of different candidates winning sum to 1.
- Insurance: probability of different claim categories.
Exam Tips
Common Mistakes
MistakeAdding probabilities for events that are NOT mutually exclusive, like King or Heart
Fixonly add probabilities directly if the events truly cannot happen at the same time
MistakeForgetting that all mutually exclusive outcomes must sum to exactly 1
Fixalways check your probabilities add up to 1 - if not, you've made an error or missed an outcome
MistakeConfusing "and" with "or" when combining mutually exclusive events
Fixmutually exclusive events use "OR" with addition; "AND" (combined) events need a different approach
MistakeMixing probabilities as percentages and decimals inconsistently in the same calculation
Fixkeep probabilities in the same format (fractions, decimals, or percentages) throughout
MistakeTrying to divide to find a missing probability instead of subtracting from 1
Fixto find a missing probability among mutually exclusive outcomes, subtract the known probabilities from 1
MistakeTreating every item in a list as equally likely without checking whether any value repeats
Fixcount how many times each outcome actually appears in the list, then divide by the total number of items
For Exams
- Check events are truly mutually exclusive before adding their probabilities.
- Remember: all possible mutually exclusive outcomes sum to exactly 1.
- To find a missing probability, subtract the known probabilities from 1.
- The addition rule extends to any number of mutually exclusive events.
- Keep a consistent format (fraction/decimal/percentage) throughout your working.
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