Introduction to Probability
Measure the likelihood of any event using the probability formula, and always get a number between 0 and 1.
Printable Worksheets
Print or save as PDF, or build a custom worksheet from any module's questions.
Why does this matter?
Weather apps say 70% chance of rain. Doctors say a treatment works 80% of the time. The lottery jackpot has a 1 in 45 million chance. All of these are probability the mathematics of uncertainty. Once you master the formula, you can calculate the likelihood of anything.
The Big Idea
Probability measures how likely an event is to occur. We calculate it using:
$$P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of equally likely outcomes}}$$
Probability is always between 0 (impossible) and 1 (certain). A result of 0.5 means equally likely to happen or not happen.
Key Terms
Spot the Trap
- Writing probability as a ratio: "3:7" is NOT probability. Probability must be a fraction, decimal, or percentage, not a ratio. P(red) = 3/10, NOT 3:7.
- Listing duplicate outcomes: When listing the sample space for a coin flip, don't write {H, H, T}, each outcome appears once: {H, T}.
- Probability greater than 1: If you calculate P > 1, you've made an error. Check your denominator is the total number of outcomes.
Explore the Concept
Listing the Sample Space
+5 XPBefore calculating probability, you must list ALL possible outcomes, the sample space. Every outcome must appear exactly once, and all outcomes must be equally likely.
Die sample space: {1, 2, 3, 4, 5, 6}, 6 equally likely outcomes
Coin sample space: {H, T}, 2 equally likely outcomes
Picking a vowel from A–E: {A, B, C, D, E}, 5 total; {A, E} are favourable
A complete sample space leaves nothing out and includes nothing twice. This is the foundation of every probability calculation.
Calculating P(event)
+5 XPOnce you have the sample space, count how many outcomes are favourable (match your event), then divide by the total number of outcomes.
$$P(\text{event}) = \frac{\text{favourable outcomes}}{\text{total outcomes}}$$
Express your answer as a fraction, then convert to a decimal or percentage if needed. Always check: is your answer between 0 and 1?
Event: Rolling an even number on a die
Favourable: {2, 4, 6} → 3 outcomes
Total: {1, 2, 3, 4, 5, 6} → 6 outcomes
$$P(\text{even}) = \frac{3}{6} = \frac{1}{2} = 0.5$$
Complementary Events
+5 XPEvery event A has a complement, the event "not A", which includes all outcomes NOT in A. Together they cover everything, so their probabilities add to 1.
$$P(\text{not } A) = 1 - P(A)$$
If P(rain) = 0.7, then P(no rain) = 1 − 0.7 = 0.3
If P(rolling 4) = 1/6, then P(not rolling 4) = 1 − 1/6 = 5/6
The complement rule is useful when the "not" event is easier to calculate than the event itself.
Worked Examples
Die, Sample Space and P(rolling 4)
+15 XPA standard die has faces: 1, 2, 3, 4, 5, 6.
Sample space S = {1, 2, 3, 4, 5, 6}6 equally likely outcomes.
Event: roll a 4. Only one outcome matches: {4}.
Favourable outcomes = 1
$$P(4) = \frac{1}{6} \approx 0.167 \approx 16.7\%$$
This makes sense, a small probability for one specific face.
Cards, P(red card)
+15 XPA standard deck has 52 cards: 26 red (hearts + diamonds) and 26 black (clubs + spades).
Total outcomes = 52
Event: pick a red card. Favourable = all hearts and diamonds.
Favourable outcomes = 26
$$P(\text{red}) = \frac{26}{52} = \frac{1}{2} = 0.5 = 50\%$$
A 50% chance makes sense, exactly half the deck is red.
Complement Rule in Action
+15 XPA bag contains 3 red, 5 blue, and 2 green marbles (10 total).
You are asked: what is P(not red)?
$$P(\text{red}) = \frac{3}{10} = 0.3$$
$$P(\text{not red}) = 1 - P(\text{red}) = 1 - 0.3 = 0.7$$
Check: 5 blue + 2 green = 7 marbles that are not red. 7/10 = 0.7 ✓
Common Pitfalls
- Duplicate outcomes in sample space: Rolling two dice, "3 then 5" is different from "5 then 3". Don't merge them.
- P greater than 1: You can never have more favourable outcomes than total outcomes. Always check.
- Non-exhaustive complementary addition: P(A) + P(B) = 1 only works when B = "not A". If there are other outcomes, you can't use this directly.
Copy-ready Notes
Probability = favourable outcomes ÷ total outcomes. Always between 0 and 1.
Sample space: Complete list of all equally likely outcomes, each listed exactly once.
Complementary events: P(not A) = 1 − P(A). Every event and its complement sum to 1.
Key checks: P must be between 0 and 1. Write as fraction first, then convert.
Brain Trainer
A die is rolled. What is P(even number)? Write as a fraction and decimal.
Even numbers on a die: {2, 4, 6} → 3 favourable outcomes out of 6 total.
$$P(\text{even}) = \frac{3}{6} = \frac{1}{2} = 0.5$$
A bag has 3 red, 5 blue, and 2 green marbles. What is P(blue)?
Favourable (blue) = 5. Total = 3 + 5 + 2 = 10.
$$P(\text{blue}) = \frac{5}{10} = \frac{1}{2} = 0.5$$
If P(A) = 0.3, what is P(not A)?
Use the complement rule: P(not A) = 1 − P(A) = 1 − 0.3 = 0.7
A student says P(winning a game) = 1.5. Explain why this is impossible.
Probability is always between 0 and 1 (inclusive). A value of 1.5 would mean more favourable outcomes than total outcomes, which is impossible. The student likely counted incorrectly, perhaps double-counting outcomes or using the wrong denominator.
Multiple Choice
1. What is the sample space for flipping a single coin?
- H
- {H, T}
- {H, H, T, T}
- {H, T, HT}
2. A fair die is rolled. What is P(rolling a 3)?
- 3
- 1/3
- 1/6
- 3/6
3. If P(A) = 0.4, what is P(not A)?
- 0.6
- 0.4
- 1.4
- 0.04
4. A letter is chosen at random from {A, B, C, D, E}. What is P(vowel)?
- 1/5
- 3/5
- 5/2
- 2/5
5. A bag has 3 red, 4 blue, and 3 green marbles. What is P(red)?
- 3/4
- 3/10
- 1/3
- 3/7
Short Answer
Q6. A bag contains 2 red, 3 blue, and 5 green counters. A counter is picked at random.
(a) Write the sample space (by colour groups).
(b) Calculate P(green).
(c) Calculate P(not green).
Q7. A letter is chosen at random from the word MATHEMATICS.
(a) Write out the sample space (with no duplicates).
(b) Find P(choosing the letter M).
(c) Find P(choosing a vowel).
Q8. A student estimates the probability that it will rain tomorrow is 0.65.
(a) What is the probability it will NOT rain tomorrow?
(b) Is the student's estimate reasonable? Explain.
Check Your SAQ Answers
Q6.
(a) Sample space: {red, red, blue, blue, blue, green, green, green, green, green} or by type: {red × 2, blue × 3, green × 5}, 10 counters total.
(b) P(green) = 5/10 = 1/2 = 0.5
(c) P(not green) = 1 − 1/2 = 1/2 = 0.5
Q7. MATHEMATICS has letters: M, A, T, H, E, M, A, T, I, C, S, 11 letters total.
(a) Unique letters (sample space): {M, A, T, H, E, I, C, S}, but for probability we use all 11 letters (not unique) as our total.
(b) Letter M appears twice: P(M) = 2/11
(c) Vowels: A, E, A, I → 4 vowels: P(vowel) = 4/11
Q8.
(a) P(no rain) = 1 − 0.65 = 0.35
(b) Yes, the estimate is reasonable. It is between 0 and 1, and 0.65 means "more likely to rain than not", a sensible estimate given many real weather situations.
Stretch, The Lucky Spinner
+25 XPA spinner is divided into 8 equal sections numbered 1 through 8.
(a) List the sample space.
(b) What is P(prime number)? [Recall: 2, 3, 5, 7 are prime.]
(c) What is P(number greater than 5)?
(d) A friend says "P(prime) + P(not prime) should equal 2 because there are two events." Explain the error.
(e) If the spinner is spun twice, can we still use the basic formula P = favourable ÷ total? Why or why not?
Lesson Summary
- Probability is always between 0 (impossible) and 1 (certain).
- Sample space: list all equally likely outcomes exactly once.
- Formula: P(event) = favourable outcomes ÷ total outcomes.
- Complement rule: P(not A) = 1 − P(A).
- Express probability as a fraction, decimal, or percentage, never as a ratio.
True or False Revisit
Probability is always a number between 0 and 1 inclusive.
P(A) + P(A) = 1 for any event A.
If P(rain) = 0.6, then P(no rain) = 0.4.
Writing probability as a ratio (e.g., 3:7) is an acceptable alternative to a fraction.