CH – 14 PROBABILITY

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Chapter Overview

In this chapter, you will learn how to measure the chance of an event happening using mathematics.

• What probability means: Understanding how to measure uncertainty using numerical values.
• Random experiments: Performing actions where the possible outcomes are known, but the exact result cannot be predicted in advance.
• Outcomes: Identifying all possible individual results of an experiment.
• Events: Defining specific collections of outcomes that we are interested in.
• Favourable outcomes: Counting the outcomes that satisfy the conditions of a given event.
• Total possible outcomes: Finding the total number of all possible results of an experiment.
• Classical probability: Calculating theoretical probability using the ratio of favourable outcomes to total outcomes under the assumption of equally likely outcomes.
• Finding probability of simple events: Solving problems based on coins, dice, marbles, cards, and daily life situations.
• Understanding probability values: Learning why probability always lies between 0 and 1, including impossible and certain events.

🎯 Board Exam Focus

Mastering the following core skills will help you solve board exam problems accurately:

• Understanding the experiment: Carefully identifying what action is being performed (tossing coins, throwing dice, drawing cards).
• Listing all possible outcomes: Writing down the complete sample space without missing any result.
• Finding favourable outcomes: Identifying only those outcomes that satisfy the event condition.
• Finding total outcomes: Counting the exact total number of all possible results.
• Writing probability correctly: Using proper notation like P(E) and writing the formula before substituting values.
• Simplifying fractions: Reducing probability fractions to their simplest form.
• Understanding impossible and certain events: Identifying when an event has a probability of 0 or 1.
• Solving questions involving coins, dice and cards: Mastering standard problems involving single or multiple coins, dice, and playing card decks.

Most Important Concepts

What is Probability?

Probability is a branch of mathematics that measures the chance of an event happening. In simple words, probability tells us how likely or unlikely something is to occur.

In daily life, we often use words like “probably,” “chance,” or “doubt.” Probability gives a numerical value to these chances.

Examples:

• The chance of getting a Head when tossing a coin.
• The chance of getting a 6 when throwing a dice.

What is a Random Experiment?

A random experiment is an activity or action where we know all the possible results in advance, but we cannot predict the exact result before performing it.

Every time the experiment is conducted under fair conditions, all outcomes have an equal chance of occurring (equally likely outcomes).

Examples:

• Tossing a fair coin.
• Throwing an unbiased six-sided dice.
• Drawing a card from a well-shuffled deck.

VISUAL REPRESENTATION 1 — RANDOM EXPERIMENT

Plaintext

       TOSS A COIN
            ↓
     Possible Result
            ↓
      HEAD or TAIL

What is an Outcome?

An outcome is a single possible result of a random experiment.

For example: When a coin is tossed once, the possible outcomes are:

• Head (H)
• Tail (T)

Each of these results is called an outcome.

VISUAL REPRESENTATION 2 — OUTCOMES OF A COIN

Plaintext

          COIN
         ↙    ↘
      HEAD    TAIL

What is an Event?

An event is a collection of one or more outcomes of a random experiment. We usually denote an event by capital letters like E, F, or A.

For example: When throwing a single dice, let E be the event of Getting an even number. The possible favourable outcomes for this event are 2, 4, and 6.

An event that has only one outcome is called an elementary event. The sum of the probabilities of all the elementary events of an experiment is always 1.

🧠 Favourable Outcomes

Favourable outcomes are the specific outcomes that satisfy the condition of the event we want to happen.

For example: When throwing a six-sided dice:

• Total outcomes = 1, 2, 3, 4, 5, 6
• Event = Getting a number greater than 4
• Favourable outcomes = 5, 6 (because only 5 and 6 are greater than 4)

Here, the number of favourable outcomes is 2.

VISUAL REPRESENTATION 3 — EVENT AND FAVOURABLE OUTCOMES

Plaintext

TOTAL OUTCOMES
1   2   3   4   5   6

EVENT: Getting an even number

FAVOURABLE OUTCOMES
    2       4       6

📌 The Main Probability Formula

The classical or theoretical probability of an event E, written as P(E), is defined as:

Probability of an Event = Number of favourable outcomes ÷ Total number of possible outcomes

In short form:

P(E) = Number of favourable outcomes ÷ Total number of possible outcomes

Where:

• P(E) means the probability of event E happening.
• Favourable outcomes are the specific results that satisfy event E.
• Total outcomes are all possible results of the experiment.

Simple Example: A bag contains 3 red balls and 5 black balls. A ball is drawn at random. Find the probability of getting a red ball.

• Total number of balls (Total outcomes) = 3 + 5 = 8
• Number of red balls (Favourable outcomes) = 3
• P(getting a red ball) = 3 ÷ 8

VISUAL REPRESENTATION 4 — PROBABILITY SOLVING FLOW

Plaintext

     UNDERSTAND THE EVENT
              ↓
  LIST ALL POSSIBLE OUTCOMES
              ↓
  FIND FAVOURABLE OUTCOMES
              ↓
         COUNT BOTH
              ↓
FAVOURABLE OUTCOMES ÷ TOTAL OUTCOMES
              ↓
      FINAL PROBABILITY

🧠 Step-by-Step Method to Solve Probability Questions

Follow these 7 simple steps to solve any probability question correctly:

1. Step 1: Understand what the question is asking. Read the question carefully to identify the experiment and the specific event.
2. Step 2: List all possible outcomes. Write down the complete list of possible results for the given experiment.
3. Step 3: Find the favourable outcomes. Identify which outcomes satisfy the given event.
4. Step 4: Count the favourable outcomes. Count how many outcomes are in your favourable list.
5. Step 5: Count the total possible outcomes. Count the total number of all possible outcomes.
6. Step 6: Use the probability formula. Put the values into: P(E) = Number of favourable outcomes ÷ Total number of possible outcomes.
7. Step 7: Simplify the answer if possible. Convert the fraction into its simplest form or decimal.

📌 Probability of a Coin Toss

When a fair coin is tossed, there are 2 equally likely outcomes:

• Head (H)
• Tail (T)

Total possible outcomes = 2

• Probability of getting a Head: Favourable outcome = H (1 outcome) P(Head) = 1 ÷ 2
• Probability of getting a Tail: Favourable outcome = T (1 outcome) P(Tail) = 1 ÷ 2

When two coins are tossed simultaneously, the possible outcomes are: (H, H), (H, T), (T, H), (T, T) Total possible outcomes = 4

• Probability of getting at least one Head: Favourable outcomes = (H, H), (H, T), (T, H) = 3 outcomes P(at least one Head) = 3 ÷ 4

VISUAL REPRESENTATION 6 — COIN PROBABILITY

Plaintext

TOTAL OUTCOMES = 2 (Head, Tail)

HEAD → 1 favourable outcome  →  P(Head) = 1 ÷ 2
TAIL → 1 favourable outcome  →  P(Tail) = 1 ÷ 2

📌 Probability of Throwing a Dice

When a standard six-sided dice is thrown once, the possible outcomes are: 1, 2, 3, 4, 5, 6 Total possible outcomes = 6

VISUAL REPRESENTATION 7 — DICE OUTCOMES

Plaintext

DICE OUTCOMES
1   2   3   4   5   6

EVENT → Getting a number greater than 4
FAVOURABLE OUTCOMES → 5, 6

Examples of Dice Events:

• Getting an even number: Favourable outcomes = 2, 4, 6 (3 outcomes) P(even number) = 3 ÷ 6 = 1 ÷ 2
• Getting an odd number: Favourable outcomes = 1, 3, 5 (3 outcomes) P(odd number) = 3 ÷ 6 = 1 ÷ 2
• Getting a number greater than 4: Favourable outcomes = 5, 6 (2 outcomes) P(number > 4) = 2 ÷ 6 = 1 ÷ 3
• Getting a prime number: Prime numbers on a dice are 2, 3, 5 (3 outcomes) P(prime number) = 3 ÷ 6 = 1 ÷ 2

📌 Probability Using Playing Cards

A standard deck of playing cards contains a total of 52 cards. The deck is divided into 4 suits of 13 cards each:

• Spades (Black) — 13 cards
• Clubs (Black) — 13 cards
• Hearts (Red) — 13 cards
• Diamonds (Red) — 13 cards

Color Distribution:

• Total Red cards: 13 Hearts + 13 Diamonds = 26 cards
• Total Black cards: 13 Spades + 13 Clubs = 26 cards

Card Types in Each Suit: Each suit has 13 cards: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King.

Face Cards: Kings, Queens, and Jacks are called Face Cards.

• 4 Kings + 4 Queens + 4 Jacks = 12 Face Cards in total.
• Red face cards = 6 (2 Kings, 2 Queens, 2 Jacks)
• Black face cards = 6 (2 Kings, 2 Queens, 2 Jacks)

Example: Find the probability of drawing an Ace from a deck of 52 cards.

• Total possible outcomes = 52
• Number of Aces (Favourable outcomes) = 4
• P(Ace) = 4 ÷ 52 = 1 ÷ 13

📌 Important Probability Values

For any event E, the probability value always satisfies:

0 ≤ P(E) ≤ 1

This means:

• Probability can never be negative (less than 0).
• Probability can never be greater than 1.

Understanding the Values:

• P(E) = 0: The event is an Impossible Event (it can never happen).
• P(E) = 1: The event is a Certain Event or Sure Event (it is guaranteed to happen).
• 0 < P(E) < 1: The event may or may not happen.

Complementary Events: The event “not E” is written as Ē (E-bar). It is the complement of event E. The sum of the probability of an event and its complement is always 1:

P(E) + P(not E) = 1 or P(E) + P(Ē) = 1

P(not E) = 1 − P(E)

VISUAL REPRESENTATION 5 — PROBABILITY SCALE

Plaintext

IMPOSSIBLE                        CERTAIN
    0 ────────────── 0.5 ────────────── 1

0 = Impossible event (P = 0)
Between 0 and 1 = Possible event
1 = Certain event (P = 1)

🧠 Impossible Event and Certain Event

Impossible Event

An event that can never happen has a probability of 0.

Example: Getting a number 8 in a single throw of a standard six-sided dice. Since a dice only has numbers 1 to 6, getting 8 is impossible. P(getting 8) = 0 ÷ 6 = 0.

Certain Event (Sure Event)

An event that is guaranteed to happen has a probability of 1.

Example: Getting a number less than 7 in a single throw of a standard six-sided dice. Since every face has a number less than 7 (1, 2, 3, 4, 5, 6), all 6 outcomes are favourable. P(getting a number < 7) = 6 ÷ 6 = 1.

🧠 How to Check Your Answer

Before finalizing your answer in the exam, ask yourself these quick sanity checks:

• Is the probability between 0 and 1? If your answer is negative or greater than 1 (like 5/4 or 1.2), it is definitely incorrect.
• Is the number of favourable outcomes correct? Double-check if you missed any outcome or counted extra ones.
• Did I count all possible outcomes? Ensure the denominator represents the total sample space.
• Did I simplify the fraction? Always reduce fractions to their lowest terms (e.g., write 3/6 as 1/2).
• Does the answer make sense? If an event is very likely, its probability should be closer to 1 than to 0.

🧠 Concepts Students Find Difficult

• Understanding favourable outcomes: Students often count total items instead of only those satisfying the event condition.
• Counting total outcomes: When two coins or two dice are rolled together, students forget outcomes like (1, 4) and (4, 1) are distinct.
• Forgetting possible outcomes: Missing 1 or 2 as prime numbers when throwing a dice. (Remember: 1 is NOT a prime number; 2 IS the smallest prime number).
• Confusing an event with an outcome: An outcome is a single result (e.g., getting 4), while an event can be a set of outcomes (e.g., getting an even number).
• Finding prime numbers on a dice: Students often mistakenly treat 1 as prime or forget that 2 is prime. Prime numbers on a dice are 2, 3, and 5.
• Simplifying probability fractions: Leaving fractions unsimplified (e.g., leaving 4/52 instead of writing 1/13).
• Understanding probability values between 0 and 1: Forgetting that probability cannot be negative or greater than 100% / 1.

⚠️ Common Mistakes Students Make

1. Counting Favourable Outcomes Incorrectly
   • Wrong: Taking 1 as a prime number on a dice, giving 4 prime numbers (1, 2, 3, 5).
   • Correction: Remember 1 is neither prime nor composite. Prime numbers on a dice are only 2, 3, and 5.
2. Forgetting Distinct Outcomes in Two-Item Experiments
   • Wrong: Treating (1, 4) and (4, 1) as the exact same outcome when throwing two dice.
   • Correction: (1, 4) means 1 on first die and 4 on second die, while (4, 1) means 4 on first die and 1 on second die. They are 2 separate outcomes.
3. Using Favourable Outcomes as the Denominator
   • Wrong: Writing P(E) = Total outcomes ÷ Favourable outcomes.
   • Correction: The total outcomes MUST always be in the denominator (bottom of the fraction).
4. Getting a Probability Greater than 1
   • Wrong: Writing P(E) = 5/4 or 1.25.
   • Correction: Probability can NEVER be greater than 1. If your numerator is larger than the denominator, re-check your counting.
5. Confusing Impossible and Unlikely Events
   • Wrong: Thinking an event with a very small probability (like 1/1000) is an impossible event.
   • Correction: An impossible event has a probability of EXACTLY 0. Unlikely events still have a positive probability greater than 0.
6. Not Simplifying Fractions
   • Wrong: Leaving the final answer as 4/52 or 2/6.
   • Correction: Always simplify fractions to lowest terms: 4/52 = 1/13 and 2/6 = 1/3.
7. Misreading What the Event Asks
   • Wrong: Finding P(getting an Ace) when the question asks for P(NOT getting an Ace).
   • Correction: Read the question carefully. If asked for “not E”, use P(not E) = 1 − P(E).

🔥 Priority Revision

⭐⭐⭐ HIGH-PRIORITY PRACTICE

• Probability formula and basic calculations for single events.
• Single and double coin toss problems.
• Single dice problems (even, odd, prime, greater than / less than numbers).
• Playing cards basic problems (suits, colors, face cards, aces).
• Complementary events using P(E) + P(Ē) = 1.

⭐⭐ IMPORTANT PRACTICE

• Two dice thrown simultaneously (finding sum of numbers, doublets).
• Bag problems with colored balls or marbles.
• Identification of impossible (P = 0) and certain (P = 1) events.
• Problems on non-defective / defective items (shirts, pens, bulbs).

⭐ ADDITIONAL REVISION

• Geometric / area-based probability problems (helicopter crash, land in circle).
• Birthday problems and game winning/losing probabilities.
• Numbered discs / cards problems (divisibility, perfect squares, two-digit numbers).

⚡ Probability Quick Revision Sheet

• Random Experiment: Action with known possible results, but uncertain exact outcome.
• Outcome: A single result of an experiment.
• Event (E): Collection of favourable outcomes.
• Favourable Outcomes: Outcomes satisfying the given event condition.
• Main Formula: P(E) = Number of favourable outcomes ÷ Total number of possible outcomes.
• Probability Range: 0 ≤ P(E) ≤ 1 for any event E.
• Impossible Event: An event that cannot happen; P(E) = 0.
• Certain Event: An event that is sure to happen; P(E) = 1.
• Complementary Event: P(not E) = 1 − P(E).
• Elementary Event Sum: Sum of probabilities of all elementary events of an experiment = 1.

📝 Test Yourself

1. A coin is tossed once. What is the probability of getting a Tail?
2. A single fair dice is thrown. Find the probability of getting a number divisible by 3.
3. A bag contains 4 red balls, 3 green balls, and 5 blue balls. A ball is drawn at random. What is the probability that the ball drawn is green?
4. If the probability of an event E happening is 0.08, what is the probability of the event ‘not E’?
5. A card is drawn at random from a well-shuffled deck of 52 playing cards. Find the probability of getting a red queen.
6. What is the probability of getting a number greater than 6 in a single throw of a standard six-sided dice?
7. In a class of 30 students, 18 are girls and 12 are boys. If one student is chosen at random as class monitor, find the probability that the selected student is a boy.
8. Two coins are tossed simultaneously. Find the probability of getting exactly two heads.
9. A box contains 20 discs numbered from 1 to 20. A disc is drawn at random from the box. Find the probability that it bears a prime number.
10. A standard six-sided dice is thrown once. What is the probability of getting a number less than or equal to 6?

🎯 Before Your Exam

☐ I understand what probability means.

☐ I understand random experiments.

☐ I know what an outcome is.

☐ I understand events.

☐ I can find favourable outcomes.

☐ I can find total possible outcomes.

☐ I know the main probability formula.

☐ I know that probability is between 0 and 1.

☐ I understand impossible events.

☐ I understand certain events.

Trust your preparation, stay confident, and approach every question step-by-step to achieve your best score!