Why probability feels like magic in everyday life
Ever wondered why weather forecasts, game odds, or even your chances of winning a raffle feel like guessing? That’s probability at work – and it shows up in every board exam question you’ll meet.
💡 In Simple Words: Probability tells you how likely something is to happen. If you have 2 red balls and 3 blue balls in a bag, the chance of pulling out a red ball is 2 out of 5, or 40%.
What is probability?
In maths, probability is a number between 0 and 1 (or 0% to 100%) that measures how often an event will occur when you repeat the same experiment many times.
Key terms you need
- Event: Anything you’re interested in – like "getting a 4 on a dice".
- Sample space: The complete list of all possible outcomes. For a single die, it’s {1,2,3,4,5,6}.
- Favourable outcomes: The outcomes that make the event happen. If the event is "even number", the favourable outcomes are {2,4,6}.
- Theoretical probability: The probability calculated using logic, not actual experiments.
- Experimental probability: The probability you get after doing the experiment a few times and counting successes.
How to calculate basic probability
The formula is super simple:
P(E) = Number of favourable outcomes ÷ Total number of outcomes
Think of it like a jar of coloured marbles. If you want the chance of picking a green marble, count how many greens there are and divide by the total marbles.
Worked example 1: Rolling a die
Question: What is the probability of getting a number greater than 4?
Step 1: List the sample space – {1,2,3,4,5,6}.
Step 2: Identify favourable outcomes – {5,6} (two numbers).
Step 3: Apply the formula: P = 2 ÷ 6 = 1/3 ≈ 0.33 (33%).
Worked example 2: Picking a card
Question: From a standard deck of 52 cards, what is the probability of drawing a king?
Step 1: Sample space = 52 cards.
Step 2: Favourable outcomes = 4 kings.
Step 3: P = 4 ÷ 52 = 1/13 ≈ 0.077 (7.7%).
Worked example 3: Two‑step experiment (independent events)
Question: A coin is tossed and a die is rolled. What is the probability of getting a head and a 3?
Since the toss and the roll don’t affect each other, multiply the individual probabilities.
P(head) = 1/2, P(3 on die) = 1/6.
Combined P = (1/2) × (1/6) = 1/12 ≈ 0.083 (8.3%).
Common tricks for board exams
- Always write down the total number of outcomes first – it saves you from missing a case.
- If the question says "without replacement", reduce the total after each draw (like drawing two cards one after another).
- For "either‑or" problems, check if events are mutually exclusive (they cannot happen together). If they are, simply add the probabilities.
- When numbers look big, simplify fractions early – it avoids messy arithmetic later.
Theoretical vs Experimental Probability
| Aspect | Theoretical | Experimental |
|---|---|---|
| How it’s found | Using logic & counting | By actually performing the experiment |
| Formula | P = favourable ÷ total | P = number of successes ÷ number of trials |
| Typical use | Exam questions, predictions | Lab work, real‑world data |
| Accuracy | Exact (if counting is right) | Depends on how many trials you do |
Quick revision summary
- Probability = favourable outcomes ÷ total outcomes.
- Sample space = list of all possible results.
- For independent events, multiply individual probabilities.
- For mutually exclusive events, add their probabilities.
- Remember to simplify fractions before writing the final answer.
📝 Likely Exam Questions
- Question: A bag contains 3 red, 4 blue and 5 green marbles. One marble is drawn at random. Find the probability that it is either red or green.
Answer: Total marbles = 12. Favourable = red (3) + green (5) = 8. P = 8/12 = 2/3. - Question: Two dice are rolled. What is the probability of getting a sum of 7?
Answer: Possible pairs that give 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) → 6 ways. Total outcomes = 6×6 = 36. P = 6/36 = 1/6. - Question: A card is drawn, replaced, and another card is drawn. Find the probability both cards are queens.
Answer: P(first queen) = 4/52 = 1/13. Since the card is replaced, the second draw is the same: 1/13. Combined P = (1/13)×(1/13) = 1/169. - Question: In a class of 30 students, 18 play cricket, 12 play football, and 5 play both. If a student is chosen at random, what is the probability that the student plays either cricket or football?
Answer: Use Inclusion‑Exclusion: P(cricket or football) = (18+12‑5)/30 = 25/30 = 5/6.