01 · Disjoint sum
A is at most 2 and B is at least 5 on one fair die roll. Find P(A∪B).
Hint
The sets {1,2} and {5,6} do not overlap.
Worked solution
2/6+2/6=4/6=2/3. Outcomes 3 and 4 are outside the union.
Understand · explore · practise
Mutually exclusive events
Recognise events that cannot occur together, use the correct addition rule, and distinguish disjoint, complementary, exhaustive and independent events.
Before you startSample spaces, unions and intersections.
01 / Can both events happen?
Mutually exclusive events share no outcomes.
Mutually exclusive means A ∩ B = ∅.
Consequently P(A ∩ B)=0, so P(A ∪ B)=P(A)+P(B).
For a single roll, even and odd cannot both occur. For two different rolls, “first roll even” and “second roll odd” can both occur: specifying the experiment matters.
Compare events on one die rollExplore
A fair six-sided die is rolled once. Select two events and inspect their shared outcomes.
Even {2,4,6} and odd {1,3,5} share no outcomes. They are mutually exclusive and their union is the whole sample space.
02 / Use the full rule first
Dropping the overlap term needs justification.
Watch: the overlap decides the addition rule
Pause, replay or seek freely. The notes explain the same idea and stay in view.
One fair die roll: A is even; B is greater than 4.Worked example
A={2,4,6}; B={5,6}; A∩B={6}
Outcome 6 satisfies both events, so they are not mutually exclusive.
P(A∪B)=3/6+2/6−1/6=4/6=2/3
Subtract the shared outcome once.
The union is {2,4,5,6}
Counting distinct outcomes confirms the result.
02 · Overlap check
A is at most 4 and B is at least 3. Find their intersection and union probabilities.
Hint
List both sets.
Worked solution
Intersection={3,4}, probability 2/6=1/3. Union={1,2,3,4,5,6}, probability 1. These events are exhaustive but not mutually exclusive.
03 / Disjoint does not always mean complementary
Complements must also cover every outcome.
A and A′ are mutually exclusive and exhaustive.
Exhaustive means their union is the whole sample space.
03 · Not complements
Why are “at most 2” and “at least 5” not complements?
Hint
Which outcomes are left out?
Worked solution
They exclude 3 and 4 from their union. The complement of at most 2 is greater than 2, namely {3,4,5,6}.
04 · Complement probability
If P(A)=0.37, find P(A′).
Hint
A and A′ partition the sample space.
Worked solution
P(A′)=1−0.37=0.63.
04 / More than two disjoint events
Pairwise exclusivity lets their probabilities add.
A fair die is classified as low {1,2}, middle {3,4}, high {5,6}.Worked example
The categories are pairwise disjoint
No result belongs to two categories.
They are exhaustive
Every possible result belongs to one category.
Each has probability 1/3 and the total is1
Together they form a partition.
05 · Missing category
Three disjoint exhaustive events have probabilities 0.18,0.47 and p. Find p.
Hint
Their probabilities sum to 1.
Worked solution
p=1−0.18−0.47=0.35.
06 · Not exhaustive
Three pairwise-disjoint events have probabilities 0.18,0.47 and 0.20. What is the probability of none?
Hint
Their union has probability 0.85.
Worked solution
P(none)=1−(0.18+0.47+0.20)=0.15.
05 / Exclusivity and independence answer different questions
Cannot happen together is different from does not change the probability.
Independence requires P(A ∩ B)=P(A)P(B).
If both probabilities are positive, mutually exclusive events cannot be independent.
On one fair die roll, even and odd each have probability 1/2.Worked example
Their intersection probability is0
They cannot occur together.
P(even)P(odd)=1/4
This is not equal to the intersection probability.
Therefore they are not independent
Knowing the result is even rules out odd.
07 · Independent overlap
Let A be even and B be a multiple of 3 on one fair die roll. Are these independent?
Hint
Compare the intersection probability with the product.
Worked solution
A={2,4,6}, B={3,6}, intersection={6}. P(A∩B)=1/6=(1/2)(1/3), so yes. They are not mutually exclusive because6 is shared.
08 · Zero-probability edge
The empty event is disjoint from every event. Can it also be independent of an event B?
Hint
Compare0 with0×P(B).
Worked solution
Yes. Both equal0. The statement that exclusive events cannot be independent needs both probabilities to be positive.
06 / Define events precisely
Conditions on different trials are different events.
09 · Two rolls
On two independent fair die rolls, A means first roll even and B means second roll odd. Are they mutually exclusive?
Hint
Give an outcome in both.
Worked solution
No: the ordered outcome (2,3) satisfies both. They are independent and P(A∩B)=1/4.
10 · Disjoint probability bounds
Can two mutually exclusive events have probabilities 0.7 and 0.5?
Hint
Their union must have probability at most 1.
Worked solution
No. Their probabilities would add to 1.2. Such events must overlap by at least 0.2.
07 / Explain the rule you choose
Check shared outcomes before adding.
11 · Is zero enough?
In a finite sample space where every outcome has positive probability, P(A∩B)=0. What follows?
Hint
A nonempty intersection would contain a positive-probability outcome.
Worked solution
The intersection must be empty, so the events are mutually exclusive. This inference uses the stated positive-probability finite setting.
12 · Exactly one for disjoint events
If A and B are mutually exclusive with probabilities 0.25 and 0.4, find the probability of exactly one.
Hint
Both is impossible.
Worked solution
Exactly one equals the union, giving 0.25+0.4=0.65.
08 / State the experiment and inspect the overlap
The special addition rule follows from disjointness.
Mutually exclusive events cannot occur together. Complementary events are also exhaustive. Independence concerns a product rule for the intersection, so positive-probability exclusive events are dependent. Use a short outcome list to test your interpretation.
Section 1 of 8 · Can both events happen?