Impossible and Sure Events
In probability, every event has a probability value between 0 and 1. The two extreme cases are impossible events (probability = 0) and sure events (probability = 1).
Understanding these boundary cases is essential for interpreting probability values and verifying calculations. If a computed probability falls outside the range [0, 1], the calculation contains an error.
These concepts form the foundation of the probability axioms studied in higher mathematics.
These concepts are the boundary conditions of probability theory. Every valid probability value must lie within the closed interval [0, 1]. Understanding these extremes helps in:
- Error checking: Any calculated probability outside [0, 1] is definitely wrong.
- Logical reasoning: Identifying events that cannot occur or must occur simplifies complex problems.
- Complementary events: P(E) + P(not E) = 1 is a direct consequence of these boundary conditions.
Recognising impossible and sure events quickly is a valuable exam skill. When a question asks "Find P(E) where E is getting a sum of 1 on two dice", immediately recognise this as impossible (P = 0) without listing the full sample space. Similarly, "P(getting a number from 1 to 6 on a die)" is immediately P = 1.
What is Impossible and Sure Events?
Definition — Impossible Event: An event that cannot occur under any circumstance. Its probability is 0.
Definition — Sure Event (Certain Event): An event that will definitely occur in every trial. Its probability is 1.
Key facts:
- For any event E: 0 ≤ P(E) ≤ 1
- P(impossible event) = 0
- P(sure event) = 1
- P(E) + P(not E) = 1 (complementary events)
- The complement of an impossible event is a sure event, and vice versa.
Probability Scale Interpretation:
- P = 0: The event is impossible. It will never occur in any trial.
- P close to 0 (e.g., 0.01): The event is very unlikely. It occurs about 1 in 100 trials.
- P = 0.25: The event occurs about 1 in 4 trials.
- P = 0.5: Even chance. The event occurs in about half the trials. Like flipping a fair coin.
- P = 0.75: The event is likely. It occurs in about 3 out of 4 trials.
- P close to 1 (e.g., 0.99): The event is very likely. It occurs in about 99 out of 100 trials.
- P = 1: The event is certain. It occurs in every trial.
The probability of any event is always a number between 0 and 1 inclusive. This is a fundamental axiom of probability theory established by Kolmogorov.
Impossible and Sure Events Formula
Probability Range:
0 ≤ P(E) ≤ 1
Special values:
- P(E) = 0 → E is an impossible event
- P(E) = 1 → E is a sure (certain) event
- 0 < P(E) < 1 → E is a possible but uncertain event
Complementary relationship:
P(E) + P(E̅) = 1
Where E̅ (or E') is the complement of E (the event "E does not occur").
Mathematical Axioms of Probability (Kolmogorov Axioms):
- Axiom 1 (Non-negativity): For any event E, P(E) ≥ 0.
- Axiom 2 (Normalisation): P(S) = 1, where S is the sample space (the sure event).
- Axiom 3 (Additivity): For mutually exclusive events A and B: P(A ∪ B) = P(A) + P(B).
From these three axioms, all properties of probability — including P(impossible event) = 0, the complement rule, and 0 ≤ P(E) ≤ 1 — can be derived mathematically.
Derived Properties:
- P(∅) = 0 (probability of impossible event/empty set is 0).
- P(E') = 1 − P(E) (complement rule).
- P(A ∪ B) = P(A) + P(B) − P(A ∩ B) (addition rule for non-exclusive events).
- If A ⊆ B, then P(A) ≤ P(B) (monotonicity).
Derivation and Proof
Why P(impossible event) = 0:
- An impossible event has no favourable outcomes in the sample space.
- P(E) = (Number of favourable outcomes) / (Total outcomes) = 0/n = 0.
Why P(sure event) = 1:
- A sure event includes all outcomes in the sample space.
- P(E) = n/n = 1.
Why 0 ≤ P(E) ≤ 1:
- The number of favourable outcomes (f) satisfies: 0 ≤ f ≤ n (total outcomes).
- Dividing: 0/n ≤ f/n ≤ n/n → 0 ≤ P(E) ≤ 1.
- P(E) cannot be negative (no negative counts of outcomes).
- P(E) cannot exceed 1 (favourable cannot exceed total).
Sum of All Probabilities = 1:
- The sample space S contains all possible outcomes.
- P(S) = n/n = 1 (all outcomes are in S).
- If E₁, E₂, ..., Eₖ are all possible mutually exclusive events that cover S, then P(E₁) + P(E₂) + ... + P(Eₖ) = 1.
- Example: For a die, P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1/6 × 6 = 1.
Complementary Events Rule Proof:
- E and E' (not E) are complementary — together they cover the entire sample space.
- E and E' are mutually exclusive — no outcome is in both.
- Therefore: P(E) + P(E') = P(S) = 1.
- Rearranging: P(E') = 1 - P(E).
Types and Properties
Classification of Events by Probability:
| P(E) | Type | Meaning | Example |
|---|---|---|---|
| 0 | Impossible | Cannot happen | Rolling a 7 on a standard die |
| 0 to 0.25 | Very unlikely | Rarely happens | Rolling a 6 twice in a row (1/36) |
| 0.25 to 0.5 | Unlikely | Less than even chance | Drawing an ace from a deck (4/52) |
| 0.5 | Even chance | Equally likely to happen or not | Getting heads on a fair coin |
| 0.5 to 0.75 | Likely | More than even chance | Rolling less than 5 on a die (4/6) |
| 0.75 to 1 | Very likely | Usually happens | Drawing a non-ace from a deck (48/52) |
| 1 | Sure/Certain | Always happens | Rolling ≤ 6 on a standard die |
Examples of Each Category:
Impossible events (P = 0):
- Rolling a number greater than 6 on a standard die.
- Drawing a green ball from a bag with only red and blue balls.
- Getting a sum of 1 when rolling two dice (minimum sum is 2).
- Getting a sum of 13 when rolling two dice (maximum sum is 12).
- A February having 30 days.
Sure events (P = 1):
- Rolling a number between 1 and 6 (inclusive) on a standard die.
- Getting heads or tails on a fair coin toss.
- Drawing a card from a standard deck that is either red or black.
- A triangle's angles summing to 180°.
- The sun rising tomorrow (practically certain, P ≈ 1).
Neither impossible nor sure (0 < P < 1):
- Getting a 6 on a die: P = 1/6.
- Drawing a king from a deck: P = 4/52 = 1/13.
- Getting at least one head in 3 coin tosses: P = 7/8.
Practice: Classify These Events:
- Getting a number between 1 and 6 on a die → Sure (P = 1)
- Getting a 7 on a die → Impossible (P = 0)
- Getting a prime number on a die → Neither (P = 3/6 = 1/2)
- Drawing a joker from a standard 52-card deck (no jokers) → Impossible (P = 0)
- Drawing a card that is red or black → Sure (P = 1)
- Getting a sum of 14 on two dice → Impossible (max sum = 12)
- Getting a positive sum on two dice → Sure (min sum = 2 > 0)
Methods
How to identify impossible and sure events:
- List the sample space of the experiment.
- Check the event against the sample space.
- If no outcome in the sample space satisfies the event → impossible (P = 0).
- If every outcome in the sample space satisfies the event → sure (P = 1).
Quick checks:
- Is the event asking for a value outside the range of possible outcomes? → Impossible.
- Is the event a tautology (always true by definition)? → Sure.
- Does the event contradict the rules of the experiment? → Impossible.
Verifying calculations:
- If P(E) < 0 → Error in calculation. Recheck.
- If P(E) > 1 → Error in calculation. Recheck.
- If P(E) + P(E̅) ≠ 1 → Error. Recheck both values.
Using Impossible and Sure Events in Problem Solving:
- If P(E) = 0, then E cannot affect any other probability calculation — it contributes nothing.
- If P(E) = 1, then E is guaranteed — you can assume it happens and focus on other events.
- The complementary events rule P(E) + P(not E) = 1 is extremely useful: if P(E) is hard to calculate directly, calculate P(not E) instead and subtract from 1.
- For "at least one" problems: P(at least one success) = 1 − P(zero successes). The "zero successes" event is often easier to count.
Decision Flowchart:
- Can the event possibly happen given the experiment? If no → P = 0 (impossible).
- Must the event necessarily happen regardless of outcome? If yes → P = 1 (sure).
- Otherwise → Calculate P using favourable/total.
Solved Examples
Example 1: Impossible Event — Die Shows 7
Problem: A standard die is rolled. Find P(getting 7).
Solution:
Sample space: {1, 2, 3, 4, 5, 6}
Favourable outcomes for "getting 7": None — 7 is not on a standard die.
P(7) = 0/6 = 0
This is an impossible event.
Example 2: Sure Event — Die Shows ≤ 6
Problem: A die is rolled. Find P(getting a number ≤ 6).
Solution:
Sample space: {1, 2, 3, 4, 5, 6}
Favourable outcomes: {1, 2, 3, 4, 5, 6} — all outcomes satisfy the condition.
P(number ≤ 6) = 6/6 = 1
This is a sure event.
Example 3: Impossible — Even Number on Odd-Numbered Spinner
Problem: A spinner has sections numbered 1, 3, 5, 7, 9. Find P(getting an even number).
Solution:
Sample space: {1, 3, 5, 7, 9} — all odd numbers.
Favourable for "even number": None.
P(even) = 0/5 = 0
Getting an even number on this spinner is impossible.
Example 4: Sure Event — Positive Number from {2, 4, 6, 8}
Problem: A number is drawn from {2, 4, 6, 8}. Find P(getting a positive number).
Solution:
Sample space: {2, 4, 6, 8} — all are positive.
Favourable: {2, 4, 6, 8} — all 4 outcomes.
P(positive) = 4/4 = 1
This is a sure event.
Example 5: Complement of an Impossible Event
Problem: If E is the event "rolling a 0 on a standard die", find P(E) and P(not E).
Solution:
P(E) = 0 (impossible — die has no 0).
P(not E) = 1 − P(E) = 1 − 0 = 1
The complement of an impossible event is a sure event.
Example 6: Drawing a Red Ball from Blue-Only Bag
Problem: A bag contains 5 blue balls. Find P(drawing a red ball).
Solution:
Sample space: 5 blue balls
Favourable (red ball): 0 — no red balls exist in the bag.
P(red) = 0/5 = 0
This is an impossible event.
Example 7: Sure Event — Drawing a Ball from a Bag
Problem: A bag has 3 red and 4 blue balls. Find P(drawing a ball that is either red or blue).
Solution:
Total balls = 7. All are either red or blue.
Favourable: Every ball is red or blue → 7 outcomes.
P(red or blue) = 7/7 = 1
This is a sure event.
Example 8: Verifying Probability Range
Problem: A student calculates P(E) = 5/3 for some event. Is this correct?
Solution:
Check: 5/3 ≈ 1.67 > 1.
Since P(E) must satisfy 0 ≤ P(E) ≤ 1, the value 5/3 is not valid.
The student's calculation contains an error. Possible mistakes:
- Favourable outcomes counted more than once.
- Total outcomes counted incorrectly (too few).
- Arithmetic error in division.
Example 9: Impossible Event on Two Dice
Problem: Two dice are thrown. Find P(sum = 1).
Solution:
Minimum sum: 1 + 1 = 2. The sum can never be 1.
Favourable outcomes: None.
P(sum = 1) = 0/36 = 0
This is an impossible event.
Example 10: Sure Event on Two Dice
Problem: Two dice are thrown. Find P(sum is between 2 and 12, inclusive).
Solution:
Range of sum: Minimum = 1 + 1 = 2. Maximum = 6 + 6 = 12.
Every possible sum lies between 2 and 12.
Favourable outcomes: All 36.
P(2 ≤ sum ≤ 12) = 36/36 = 1
This is a sure event.
Example 11: Probability Scale Classification
Problem: Classify each event as impossible, unlikely, even chance, likely, or certain:
- (a) Getting a 13 on a standard die
- (b) Getting heads on a coin
- (c) Drawing a red card from a standard deck
- (d) Getting a prime number on a die
- (e) A square having 4 sides
Solution:
- (a) P = 0 → Impossible (die only has 1-6)
- (b) P = 1/2 → Even chance
- (c) P = 26/52 = 1/2 → Even chance
- (d) P = 3/6 = 1/2 (primes: 2, 3, 5) → Even chance
- (e) P = 1 → Certain (by definition, all squares have 4 sides)
Real-World Applications
Risk Assessment:
- Insurance companies identify impossible risks (P = 0) to exclude from policies, and near-certain events (P ≈ 1) to factor into premiums.
Error Detection:
- If a probability calculation yields a value outside [0, 1], it flags an error in the model or computation.
Logic and Computing:
- Boolean logic uses 0 (false, impossible) and 1 (true, certain). Probability extends this to continuous values between 0 and 1.
Medical Testing:
- A test with 0% false negative rate has P(missing the disease) = 0 — a sure detection.
- No real-world test achieves P = 0 or P = 1 exactly, but these bounds guide design goals.
Weather Forecasting:
- A 0% chance of rain means it is impossible (based on current models). A 100% chance means it is certain. Real forecasts are always between these extremes.
Quality Control:
- A manufacturer with P(defect) = 0 has perfect quality (impossible in practice but a theoretical goal).
- P(at least one defect in 1000 items) is almost 1 for any non-zero defect rate — a near-sure event.
Sports:
- P(a team scoring negative points) = 0 (impossible by game rules).
- P(at least one team winning in a match) = 1 (sure event, ignoring draws).
Everyday Language vs Mathematical Precision:
- People say "impossible" loosely ("It's impossible to pass this exam!") — this is not mathematical impossibility.
- In mathematics, P = 0 means no outcome in the sample space satisfies the condition — a strict definition.
- Similarly, "certain" in everyday language may mean "very likely," but in mathematics, P = 1 means the event includes every outcome in the sample space.
Key Points to Remember
- Impossible event: Cannot occur. P = 0. Has no favourable outcomes.
- Sure event: Always occurs. P = 1. All outcomes are favourable.
- For any event E: 0 ≤ P(E) ≤ 1.
- P(E) + P(not E) = 1 always. This is the complementary rule.
- The complement of an impossible event is a sure event.
- The complement of a sure event is an impossible event.
- A probability of 0 does not mean "never in practice" — it means never within the defined sample space.
- If a calculation gives P < 0 or P > 1, there is definitely an error.
- Probability = 0.5 means an event is equally likely to occur or not occur.
- These concepts are the axioms of probability and apply to all probability problems.
- The sum of probabilities of all mutually exclusive and exhaustive events in a sample space is always 1.
- The complement rule P(E) + P(not E) = 1 is one of the most useful tools in probability — it lets you calculate hard probabilities by computing their complements.
- A probability value acts as a measure of likelihood on a scale from 0 (impossible) to 1 (certain).
Practice Problems
- A die is rolled. Find P(getting a number > 6).
- A die is rolled. Find P(getting a positive integer less than 7).
- A bag has 4 red balls. Find P(drawing a green ball).
- A coin is tossed. Find P(getting either heads or tails).
- Two dice are thrown. Find P(sum = 13).
- A card is drawn from a standard deck. Find P(it is a card). What type of event is this?
- Three coins are tossed. Find P(at least 0 heads). Classify this event.
- If P(E) = 0.3, find P(not E). Can P(E) be 1.2? Why?
Frequently Asked Questions
Q1. What is an impossible event?
An impossible event is one that cannot occur in any trial of the experiment. Its probability is exactly 0. Example: rolling a 7 on a standard six-faced die.
Q2. What is a sure event?
A sure event (or certain event) is one that will definitely occur in every trial. Its probability is exactly 1. Example: getting a number ≤ 6 when rolling a standard die.
Q3. Can probability be negative?
No. Probability is defined as the ratio of favourable outcomes to total outcomes. Since both are non-negative counts and total is always positive, P(E) is always between 0 and 1, inclusive.
Q4. Can probability be greater than 1?
No. The number of favourable outcomes can never exceed the total number of outcomes. So P(E) = favourable/total ≤ 1. A value greater than 1 indicates a calculation error.
Q5. What is the complement of an impossible event?
The complement of an impossible event is a sure event. If P(E) = 0, then P(not E) = 1 − 0 = 1. This makes sense: if E can never happen, then 'not E' always happens.
Q6. Is P(E) = 0 the same as 'E never happens in real life'?
In Class 10 (theoretical probability), P(E) = 0 means E has no favourable outcomes in the sample space. In advanced probability (continuous distributions), P = 0 events can technically occur but have zero likelihood. For Class 10, P = 0 means impossible.
Q7. How do I verify my probability answer is correct?
Check three things: (1) 0 ≤ P(E) ≤ 1, (2) P(E) + P(not E) = 1, (3) The sum of probabilities of all outcomes in the sample space = 1.
Q8. What are the probability axioms?
The three axioms (Kolmogorov) are: (1) P(E) ≥ 0 for any event E, (2) P(sample space) = 1, (3) For mutually exclusive events, P(A or B) = P(A) + P(B). These ensure probability stays in [0, 1].
Q9. Are impossible and sure events tested in board exams?
Yes. Questions often ask to classify events as impossible, sure, or neither. Also, questions ask to verify whether a given probability value is valid (must be between 0 and 1). These carry 1-2 marks.
Q10. Can an event have probability exactly 0.5?
Yes. P(E) = 0.5 means the event is equally likely to occur or not occur. Example: getting heads on a fair coin. This is neither impossible nor sure — it is an uncertain event with equal chances.
