L31 — Introduction to Probability
Key Terms
- Sample space S
- The set of all possible outcomes of a random experiment.
- Event
- A subset of the sample space — one or more outcomes grouped together.
- Probability P(A)
- A number between 0 and 1 measuring how likely event A is; P(A) = favourable outcomes ÷ total equally likely outcomes.
- Complement A′
- The event "A does not occur"; P(A′) = 1 − P(A).
- Mutually exclusive
- Two events that cannot both occur at the same time; A ∩ B = ∅ so P(A ∩ B) = 0.
- Addition rule
- P(A ∪ B) = P(A) + P(B) − P(A ∩ B); subtract the intersection to avoid counting it twice.
Probability basics
| Concept | Definition / Formula |
|---|---|
| Probability | P(A) = (favourable outcomes) ÷ (total equally likely outcomes) |
| Range | 0 ≤ P(A) ≤ 1 |
| Complement | P(A′) = 1 − P(A) |
| Certain event | P(A) = 1 |
| Impossible event | P(A) = 0 |
Addition rule
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
For mutually exclusive events (A and B cannot both occur): P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).
Multiplication rule (independent events)
If A and B are independent (one does not affect the other):
P(A ∩ B) = P(A) × P(B)
Key vocabulary
| Term | Meaning |
|---|---|
| Sample space S | Set of all possible outcomes |
| Event | A subset of the sample space |
| Mutually exclusive | Events that cannot both occur: A ∩ B = ∅ |
| Exhaustive | Events that cover all possibilities: P(A ∪ B ∪ …) = 1 |
| Experimental probability | Relative frequency from trials: P ≈ frequency ÷ n |
Worked Example 1 — Basic probability
A bag contains 4 red, 3 blue, and 5 green marbles. One marble is drawn at random. Find P(red) and P(not green).
Total = 12. P(red) = 4/12 = 1/3.
P(not green) = 1 − P(green) = 1 − 5/12 = 7/12.
Worked Example 2 — Mutually exclusive events
A card is drawn from a standard deck (52 cards). Find P(King or Queen).
King and Queen are mutually exclusive (a card cannot be both).
P(K or Q) = P(K) + P(Q) = 4/52 + 4/52 = 8/52 = 2/13.
Worked Example 3 — Addition rule (non-mutually exclusive)
P(A) = 0.4, P(B) = 0.5, P(A ∩ B) = 0.2. Find P(A ∪ B).
P(A ∪ B) = 0.4 + 0.5 − 0.2 = 0.7.
Worked Example 4 — Two-way table
100 students: sport (Yes/No) × music (Yes/No). Sport & Music = 18, Sport only = 32, Music only = 24, Neither = 26.
P(Sport) = 50/100 = 0.5. P(Music) = 42/100 = 0.42. P(Sport ∪ Music) = (50+42−18)/100 = 74/100 = 0.74.
Worked Example 5 — Experimental probability
A drawing pin is tossed 200 times and lands point-up 74 times. Estimate P(point-up).
P(point-up) ≈ 74/200 = 37/100 = 0.37.
As the number of trials increases, experimental probability approaches the theoretical probability (Law of Large Numbers).
-
Basic probability. Fluency
- (a) A die is rolled. Find P(even number).
- (b) A bag has 5 red and 7 blue balls. Find P(red).
- (c) P(A) = 0.35. Find P(A′).
- (d) A letter is chosen at random from the word PROBABILITY. Find P(vowel).
-
Mutually exclusive events. Fluency
- (a) P(A) = 0.3, P(B) = 0.45, A and B mutually exclusive. Find P(A or B).
- (b) A card is drawn from a deck. Find P(Ace or King).
- (c) A die is rolled. Find P(1 or 6).
- (d) P(A) = 2/7 and P(B) = 3/7, mutually exclusive. Find P(neither A nor B).
-
Addition rule. Fluency
- (a) P(A) = 0.5, P(B) = 0.4, P(A ∩ B) = 0.15. Find P(A ∪ B).
- (b) From a deck: Find P(red or face card). [Face cards: J, Q, K. Red face cards = 6.]
- (c) P(A ∪ B) = 0.8, P(A) = 0.5, P(B) = 0.4. Find P(A ∩ B).
- (d) P(A) = 0.6, P(B) = 0.5, P(A ∩ B) = 0.3. Find P(A′ ∩ B′).
-
Sample spaces. Fluency
- (a) Two coins are tossed. List the sample space and find P(exactly one head).
- (b) A die is rolled twice. How many outcomes are in the sample space?
- (c) A card is drawn from a standard 52-card deck. Find P(spade or ace).
- (d) A spinner has 8 equal sections numbered 1–8. Find P(prime number).
-
Venn diagram probabilities. Understanding
In a class of 40 students, 22 play football (F) and 18 play basketball (B). 8 students play both sports. The rest play neither.
- (a) Find P(F ∩ B) — probability of playing both sports.
- (b) Find P(F only) — plays football but not basketball.
- (c) Find P(F ∪ B) — plays at least one sport.
- (d) Find P(neither) — plays neither sport.
-
Two-way tables and probability. Understanding
200 people surveyed on coffee (Yes/No) and tea (Yes/No):
Tea Yes Tea No Total Coffee Yes 50 70 120 Coffee No 40 40 80 Total 90 110 200 - (a) Find P(Coffee Yes).
- (b) Find P(Coffee and Tea).
- (c) Find P(Coffee or Tea).
- (d) Are Coffee and Tea mutually exclusive? How can you tell?
-
Experimental probability. Understanding
- (a) A biased coin is flipped 500 times. Heads appears 310 times. Estimate P(Heads).
- (b) A quality control inspector finds 12 defective items in 300. Estimate P(defective).
- (c) As the number of trials increases, what happens to experimental probability?
- (d) A die is rolled 60 times. Each face should appear 10 times theoretically. The face “6” appears 18 times. Does this prove the die is biased? Explain.
-
Tree diagrams and counting. Understanding
A bag contains 3 red (R) and 2 blue (B) balls. Two balls are drawn without replacement.
- (a) Draw a tree diagram showing all possible outcomes.
- (b) Find P(both red).
- (c) Find P(one red and one blue).
- (d) Find P(at least one blue).
-
Card game probability. Problem Solving
Three cards are drawn one at a time without replacement from a standard 52-card deck.
- (a) Find P(all three are aces).
- (b) Find P(first card is an ace).
- (c) Find P(at least one ace in three draws).
- (d) Find P(three cards of the same suit — all spades or all hearts or all diamonds or all clubs).
-
Faulty product analysis. Problem Solving
A factory produces light bulbs. P(defective) = 0.03. Three bulbs are selected at random (independently).
- (a) Find P(first bulb is defective).
- (b) Find P(all three are defective).
- (c) Find P(none are defective).
- (d) Find P(at least one is defective).