Probability – Beginner Study Notes
For the ATMA Quantitative Ability exam
What we’ll learn
- What probability means.
- How to count favourable & total outcomes.
- Simple rules that let us add or multiply probabilities.
- A few quick tricks for solving questions.
1. What is Probability?
- Probability = “how likely something is to happen.”
- It’s a number between 0 (never) and 1 (always).
- Think of it as the share of all possible results that are good for you.
Everyday analogy
If you have a bag of 10 marbles, 3 are red.
The probability of pulling a red marble is 3/10 = 0.3 (30 %).
2. Counting Outcomes
| Term | Meaning | Example |
|---|---|---|
| Total outcomes | All ways a situation can happen | 6 ways to roll a single die (1–6) |
| Favourable outcomes | Ways the event you want occurs | 3 ways to roll an even number (2,4,6) |
Probability formula
[
P(\text{event}) = \frac{\text{favourable outcomes}}{\text{total outcomes}}
]
Tiny example
Rolling a die:
Total = 6, Favourable (rolling 5) = 1
(P(5) = 1/6)
3. Simple Rules
3.1 Addition Rule (Mutually Exclusive Events)
- Two events can’t happen at the same time.
- Add their probabilities.
[ P(A \text{ or } B) = P(A) + P(B) ]
Example
Toss a coin:
Heads probability = 1/2, Tails probability = 1/2.
Probability of getting heads or tails = 1/2 + 1/2 = 1.
3.2 Multiplication Rule (Independent Events)
- One event doesn’t affect the other.
- Multiply their probabilities.
[ P(A \text{ and } B) = P(A) \times P(B) ]
Example
Toss a coin and roll a die (they’re independent).
Probability of heads and rolling a 4 = (1/2) × (1/6) = 1/12.
3.3 Complement Rule
- Probability of “not A” = 1 – probability of A.
[ P(\text{not } A) = 1 - P(A) ]
Example
Probability of not rolling a 6 on a die = 1 – 1/6 = 5/6.
4. Quick Tips for the Exam
| Tip | Why It Helps |
|---|---|
| Write the event in words first. | Avoids confusion. |
| Draw a small table if many outcomes. | Makes counting visible. |
| Check if events are independent or exclusive. | Chooses the right rule. |
| Use the complement rule when it’s easier. | Saves time. |
5. Mini Practice Questions
| # | Question | Hint | Answer |
|---|---|---|---|
| 1 | A bag has 4 red and 6 blue marbles. Pull one at random. What is the probability it’s red? | Count total marbles. | (4/10 = 0.4) |
| 2 | Toss a fair coin twice. What is the probability of getting at least one head? | Use complement: “no heads” = tails both times. | (1 - (1/2 \times 1/2) = 3/4) |
| 3 | Roll two dice. What is the probability that the sum is 7? | List all pairs that give 7. | 6 favourable out of 36 → (6/36 = 1/6) |
| 4 | A student flips a coin and rolls a die simultaneously. What is the probability of tails and a number greater than 4? | Independent events. | ((1/2) \times (2/6) = 1/6) |
6. Key Takeaways
- Probability = favourable / total.
- Use addition for events that cannot happen together.
- Use multiplication for independent events.
- The complement trick is handy for “at least” or “not” questions.
- Keep calculations simple; write every step clearly.
Final Thought
Think of probability like sharing a pizza: every slice is a possible outcome, and the slices you actually eat are the favourable ones. The easier you see the share, the easier the math!