Ever wondered why flipping a coin feels like a 50‑50 gamble? That simple chance is the heart of probability, a tool that shows up in every board exam question.
💡 In Simple Words: Probability tells us how likely something is to happen. If you have 2 red marbles and 2 blue marbles in a bag, the chance of pulling a red one is 2 out of 4, or ½.
What is Probability?
In maths, a probability is a number between 0 and 1 that measures how often an event occurs when you repeat an experiment many times. An experiment is any action you can repeat, like rolling a dice. Each possible result is an outcome. All outcomes together form the sample space. An event is a set of outcomes you care about, such as “getting an even number”. The favorable outcomes are the ones that make the event true.
Key Terms You Must Know
- Sample space (S): the complete list of all possible outcomes. Think of it as a menu that shows every dish you could order.
- Event (E): a subset of the sample space. Like picking just the desserts from the menu.
- Favorable outcomes: the outcomes that belong to the event.
- Probability formula: P(E) = Number of favorable outcomes ÷ Total number of outcomes.
How to Solve Basic Probability Problems
Most board‑exam questions follow a simple four‑step recipe. Follow it, and you’ll rarely get stuck.
Let’s break down each step with everyday language.
Step 1 – Read the question
Know exactly what the question asks. Is it “probability of drawing a king” or “probability of getting a sum of 7”?
Step 2 – Identify total outcomes
List every possible result. For a single dice, there are 6 faces, so total outcomes = 6.
Step 3 – Count favorable outcomes
Pick the outcomes that satisfy the event. If the event is “even number”, the favorable faces are 2, 4, 6 → 3 outcomes.
Step 4 – Apply the formula
Plug the numbers into P = favorable ÷ total. Then, if you can, reduce the fraction.
Worked Example 1: Simple Dice
Question: A fair dice is rolled once. What is the probability of obtaining a number greater than 4?
Solution:
- Total outcomes = 6 (1,2,3,4,5,6).
- Favorable outcomes = {5,6} → 2 outcomes.
- Probability = 2 ÷ 6 = 1/3.
Notice how we didn’t need any complicated counting – just a quick scan of the numbers.
Worked Example 2: Drawing Cards
Question: From a standard deck of 52 cards, one card is drawn at random. Find the probability that the card is a heart or a king.
Solution:
- Total outcomes = 52.
- Hearts = 13 cards.
- Kings = 4 cards, but one of them (the King of hearts) is already counted in the hearts.
- Favorable outcomes = 13 + 4 – 1 = 16.
- Probability = 16 ÷ 52 = 4/13 after simplifying.
This example shows why it’s important to avoid double‑counting.
Quick Reference Table
| Situation | Total Outcomes (N) | Favorable Outcomes (F) | Probability (P) |
|---|---|---|---|
| Single die, even number | 6 | 3 | 1/2 |
| Two coins, both heads | 4 (HH, HT, TH, TT) | 1 (HH) | 1/4 |
| Bag with 5 red, 3 blue marbles – pick red | 8 | 5 | 5/8 |
Common Mistakes to Avoid
- Forgetting to simplify: 2/8 is still a valid probability, but 1/4 is cleaner and scores better.
- Counting the same outcome twice: The “heart or king” problem is a classic trap.
- Mixing up order when using permutations: In basic probability we usually treat outcomes as equally likely; permutations matter only when order is important.
📝 Likely Exam Questions
- Question: A box contains 4 green, 5 yellow, and 6 red balls. One ball is drawn at random. What is the probability that the ball is not yellow?
Answer: Total = 15. Non‑yellow balls = 4 + 6 = 10. Probability = 10/15 = 2/3. - Question: Two dice are rolled. Find the probability that the sum of the numbers is 9.
Answer: Possible pairs that give 9: (3,6), (4,5), (5,4), (6,3) → 4 outcomes. Total outcomes = 6×6 = 36. Probability = 4/36 = 1/9. - Question: From a deck of 52 cards, two cards are drawn one after the other without replacement. What is the probability that both are aces?
Answer: First ace: 4/52. Second ace: 3/51. Multiply: (4/52)×(3/51)=12/2652=1/221. - Question: A coin is tossed three times. What is the probability of getting exactly two heads?
Answer: Number of ways to choose 2 heads out of 3 tosses = C(3,2)=3. Total possible outcomes = 2³=8. Probability = 3/8. - Question: A bag has 3 white, 2 black, and 5 red marbles. Two marbles are drawn together. Find the probability that both are of different colours.
Answer: Total ways to choose 2 marbles = C(10,2)=45. Ways to pick same colour: white C(3,2)=3, black C(2,2)=1, red C(5,2)=10 → total same =14. Different colours = 45‑14=31. Probability = 31/45.