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Unit: Statistics
Chapter: Introduction, Sampling & Probability
Reference: – Introduction to Chance and Probability, Probability Scale (0 to 1), Impossible and Certain Events, Favourable Outcomes and Total Outcomes, Formula for Probability, Experimental Probability vs Theoretical Probability, Complement of an Event, Simple Probability Problems, Probability with Dice, Coins, and Spinners, Solved Examples, Odd-One-Out Problems, Common Mistakes
After studying this chapter, you should be able to understand:
- What is Probability and How It Measures Chance
- The Probability Scale from 0 to 1
- How to Calculate Probability as Favourable Outcomes over Total Outcomes
- Difference Between Experimental and Theoretical Probability
- The Complement of an Event
Introduction to Chance and Probability
Definition
Probability is the measure of how likely an event is to happen. It is a number between 0 and 1 (inclusive) that tells us the chance of something occurring. A probability of 0 means the event is impossible. A probability of 1 means the event is certain. Probabilities closer to 0 are unlikely; probabilities closer to 1 are likely.
When we study probability, we essentially ask:
"How likely is this event to happen? What is the chance?"
Even though we cannot predict individual random events with certainty, probability helps us understand what to expect over many trials.
Importance of Probability
- Used in weather forecasting (chance of rain)
- Used in games of chance (cards, dice, lotteries)
- Essential for risk assessment in insurance, finance, and medicine
- Helps make decisions under uncertainty
- Foundation for statistics and data science
Example
When you flip a fair coin, the chance of getting heads is 1 out of 2, or 0.5, or 50%. This does not mean every two flips gives exactly one head, but over many flips, about half will be heads.
Subtopics
1. The Probability Scale
Probabilities are always between 0 and 1. They can be written as fractions, decimals, or percentages.
Probability Scale:
0 (0%) → Impossible
0.25 (25%) → Unlikely
0.5 (50%) → Equally likely (even chance)
0.75 (75%) → Likely
1 (100%) → Certain
Examples on the Scale:
Impossible event (0): Rolling a 7 on a standard six-sided die
Unlikely event (close to 0): Winning the lottery
Even chance (0.5): Getting heads on a coin flip
Likely event (close to 1): It will rain in the desert sometime this year
Certain event (1): The sun will rise tomorrow
2. Basic Probability Formula
The probability of an event happening is:
P(event) = (Number of favorable outcomes) / (Total number of possible outcomes)
All outcomes must be equally likely for this formula to work.
Example 1 – Rolling a die:
What is the probability of rolling a 4 on a fair six-sided die?
Favourable outcomes: {4} → 1 outcome
Total possible outcomes: {1, 2, 3, 4, 5, 6} → 6 outcomes
P(4) = 1/6
Example 2 – Drawing a card:
What is the probability of drawing a heart from a standard deck of 52 cards?
Favourable outcomes: 13 hearts
Total outcomes: 52 cards
P(heart) = 13/52 = 1/4 = 0.25
Example 3 – Picking a marble:
A bag has 3 red marbles, 5 blue marbles, and 2 green marbles. What is the probability of picking a blue marble?
Favourable outcomes: 5 blue marbles
Total outcomes: 3 + 5 + 2 = 10 marbles
P(blue) = 5/10 = 1/2 = 0.5
3. Favourable Outcomes and Total Outcomes
Favorable outcomes are the outcomes that match the event we are interested in.
Total possible outcomes is the total number of outcomes that could possibly happen.
Both must be counted correctly.
Important: Outcomes must be equally likely. For example, when rolling a fair die, each face is equally likely. When using a spinner, each section must be the same size for equal likelihood.
4. Probability of the Complement
The complement of an event is the event NOT happening.
Complement Rule: P(not event) = 1 – P(event)
Example: If the probability of rain tomorrow is 0.3, then the probability of no rain is 1 – 0.3 = 0.7.
Why complement is useful: Sometimes it is easier to find the probability of an event NOT happening and subtract from 1.
Example – Complement method:
What is the probability of rolling at least one 6 when rolling a die twice?
Direct calculation is more complex. Instead, find probability of NO six in two rolls and subtract from 1.
P(no six in one roll) = 5/6
P(no six in two rolls) = (5/6) × (5/6) = 25/36
P(at least one six) = 1 – 25/36 = 11/36
5. Experimental vs Theoretical Probability
Theoretical Probability: What should happen based on reasoning and mathematics. It is calculated using the formula without performing any experiment.
Example: The theoretical probability of heads on a coin is 1/2.
Experimental Probability (or Relative Frequency): What actually happens when we perform an experiment. It is calculated as:
Experimental Probability = (Number of times event occurs) / (Total number of trials)
Example: You flip a coin 100 times and get heads 48 times. Experimental probability of heads is 48/100 = 0.48.
Key Relationship: As the number of trials increases, experimental probability tends to get closer to theoretical probability. This is called the Law of Large Numbers.
Example – Comparing theoretical and experimental:
A die is rolled 60 times. Theoretically, each number should appear about 10 times (60 × 1/6 = 10). Experimentally, you might get 8 twos, 12 fives, etc. The more rolls you do, the closer the experimental results get to the theoretical expectations.
6. Simple Probability Problems with Common Items
Probability with Dice:
A standard die has numbers 1 through 6.
P(even number) = 3/6 = 1/2 (even numbers: 2, 4, 6)
P(odd number) = 3/6 = 1/2 (odd numbers: 1, 3, 5)
P(number greater than 4) = 2/6 = 1/3 (numbers: 5, 6)
P(prime number) = 3/6 = 1/2 (primes: 2, 3, 5 – note 1 is not prime)
Probability with Coins:
A fair coin has heads (H) and tails (T).
P(H) = 1/2, P(T) = 1/2
With two coins, possible outcomes: HH, HT, TH, TT (4 outcomes)
P(both heads) = 1/4
P(one head, one tail) = 2/4 = 1/2 (HT and TH)
Probability with Spinners:
If a spinner has equal sections, count the favorable sections over total sections.
Example: A spinner has 4 equal colored sections: red, blue, green, yellow.
P(red) = 1/4
P(not blue) = 3/4
Probability with Bags of Marbles:
Count favorable marbles over total marbles.
Example: Bag has 4 red, 5 blue, 1 green.
P(red) = 4/10 = 2/5
P(blue or green) = (5+1)/10 = 6/10 = 3/5
P(not red) = 1 – 4/10 = 6/10 = 3/5
Solved Examples
Example 1: A fair six-sided die is rolled once. Find the probability of rolling a number less than 5.
Solution: Numbers less than 5: {1, 2, 3, 4} → 4 favorable outcomes
Total outcomes: 6
P(less than 5) = 4/6 = 2/3
Answer: 2/3
Example 2: A bag contains 7 red marbles, 3 blue marbles, and 5 green marbles. What is the probability of picking a marble that is not blue?
Solution: Total marbles = 7 + 3 + 5 = 15
Marbles that are not blue = red + green = 7 + 5 = 12
P(not blue) = 12/15 = 4/5
Answer: 4/5
Example 3: A coin is flipped three times. What is the probability of getting exactly two heads?
Solution: Total outcomes: 2 × 2 × 2 = 8 (HHH, HHT, HTH, HTT, THH, THT, TTH, TTT)
Favourable outcomes (exactly two heads): HHT, HTH, THH → 3 outcomes
P(exactly two heads) = 3/8
Answer: 3/8
Example 4: A spinner has 6 equal sections numbered 1 to 6. What is the probability of spinning a multiple of 3?
Solution: Multiples of 3 between 1 and 6: {3, 6} → 2 favourable outcomes
Total outcomes: 6
P(multiple of 3) = 2/6 = 1/3
Answer: 1/3
Example 5: In a class of 25 students, 10 have brown eyes, 8 have blue eyes, and 7 have green eyes. If one student is chosen at random, what is the probability the student does NOT have brown eyes?
Solution: Students without brown eyes = blue + green = 8 + 7 = 15
Total students = 25
P(not brown) = 15/25 = 3/5
Answer: 3/5
Common Mistakes to Avoid
Mistake 1 – Counting outcomes that are not equally likely
Not all outcomes are equally likely in some situations. Example: When rolling two dice, the sum 7 is more likely than sum 2.
Correct understanding: List outcomes carefully or use systematic counting.
Mistake 2 – Forgetting to simplify fractions
P(event) = 4/8 should be simplified to 1/2.
Correct understanding: Always simplify probability fractions to lowest terms.
Mistake 3 – Confusing "and" with "or"
"And" usually means multiply probabilities (for independent events). "Or" usually means add probabilities (for mutually exclusive events).
Correct understanding: Read problems carefully to know which operation to use.
Mistake 4 – Thinking experimental probability is always exactly theoretical
Even with a fair coin, 10 flips might give 6 heads, not 5.
Correct understanding: Experimental probability approximates theoretical probability over many trials, but may differ in small samples.
Mistake 5 – Ignoring the complement when it is easier
Finding P(at least one) directly can be hard; using 1 – P(none) is often much easier.
Correct understanding: Always consider whether the complement makes the calculation simpler.
Mistake 6 – Thinking probability 0 means impossible (true) but probability 1 means certain (also true) – forgetting that theoretical probability 0 or 1 is rare in real life
In theory, a continuous spinner landing exactly on a line has probability 0, but it could happen.
Correct understanding: In theoretical probability with finite outcomes, 0 means impossible and 1 means certain. In continuous cases, 0 can mean "almost never" not "strictly impossible."
Quick Reference Summary
Probability Formula: P(event) = (Favorable outcomes) / (Total possible outcomes)
Probability Scale: 0 (impossible) to 1 (certain)
Complement Rule: P(not event) = 1 – P(event)
Theoretical Probability: Based on reasoning, no experiment needed
Experimental Probability: Based on actual trials = (successful trials)/(total trials)
Law of Large Numbers: Experimental probability gets closer to theoretical probability as number of trials increases
Common Items:
Dice (6 faces)
Coins (2 sides)
Cards (52 cards, 4 suits, 26 red, 26 black)
Spinners (depends on number of equal sections)
Marbles in bag (count each marble)
Probability is always between 0 and 1. It can be written as fraction, decimal, or percentage.
