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Basic Probability Definition

Introduction

Probability measures how likely an event is to occur. It forms the foundation for understanding all higher-level probability concepts such as conditional probability and Bayes’ theorem. Learning the basic definition helps you calculate the likelihood of simple events like coin tosses, dice throws, or drawing a card.

Pattern: Basic Probability Definition

Pattern

The probability of an event is the ratio of the number of favourable outcomes to the total number of possible outcomes.

Formula: P(E) = Favourable Outcomes / Total Outcomes

Probability values always lie between 0 and 1.

Step-by-Step Example

Question

A coin is tossed once. Find the probability of getting (i) a head, (ii) a tail.

Solution

  1. Step 1: Identify all possible outcomes

    When a coin is tossed, there are 2 possible outcomes - Head (H) or Tail (T).

  2. Step 2: Count total outcomes

    The total number of outcomes = 2.

  3. Step 3: Count favourable outcomes

    The number of favourable outcomes for getting a Head = 1 (only H).

  4. Step 4: Compute probability

    Hence, P(Head) = 1/2 and P(Tail) = 1/2.

  5. Final Answer:

    Probability of Head = 1/2; Probability of Tail = 1/2.

  6. Quick Check:

    Total probability = 1/2 + 1/2 = 1 ✅ (correct, since total probability must equal 1).

Quick Variations

1. Tossing a die instead of a coin (find probability of getting an even number).

2. Drawing a ball from a bag with different colors (find probability of picking red).

3. Multiple coins or dice - probability of combined outcomes.

Trick to Always Use

  • Step 1: Identify total possible outcomes clearly.
  • Step 2: Count only favourable outcomes for the asked event.
  • Step 3: Apply the formula P(E) = Favourable / Total.

Summary

Summary

In the Basic Probability Definition pattern:

  • Probability = Favourable / Total.
  • All probabilities lie between 0 and 1.
  • Sum of probabilities of all possible outcomes = 1.
  • Useful for simple, direct experiments like coin, dice, and card-based events.

Practice

(1/5)
1. A fair coin is tossed once. What is the probability of getting a Head?
easy
A. 1/2
B. 1/3
C. 1
D. 0

Solution

  1. Step 1: Identify total outcomes

    For a single coin toss the sample space is {H, T} → total outcomes = 2.

  2. Step 2: Identify favourable outcomes

    Favourable outcome for Head = {H} → count = 1.

  3. Step 3: Apply formula

    P(Head) = Favourable / Total = 1/2.

  4. Final Answer:

    1/2 → Option A.

  5. Quick Check:

    P(Head) + P(Tail) = 1/2 + 1/2 = 1 ✅
Hint: Count outcomes in the sample space first, then count favourable ones.
Common Mistakes: Forgetting that a fair coin has exactly 2 equally likely outcomes.
2. A fair six-sided die is rolled once. What is the probability of getting an even number?
easy
A. 1/2
B. 1/3
C. 1/6
D. 2/3

Solution

  1. Step 1: Identify total outcomes

    A die has faces {1,2,3,4,5,6} → total outcomes = 6.

  2. Step 2: Identify favourable outcomes

    Even numbers = {2,4,6} → favourable count = 3.

  3. Step 3: Apply formula

    P(even) = 3 / 6 = 1/2.

  4. Final Answer:

    1/2 → Option A.

  5. Quick Check:

    There are 3 even and 3 odd faces, so probability of even = 3/6 = 1/2 ✅
Hint: List outcomes quickly: count evens vs total faces.
Common Mistakes: Counting only 2 even numbers (forgetting 6) or dividing incorrectly.
3. One card is drawn at random from a standard 52-card deck. What is the probability that the card is a Heart?
easy
A. 1/2
B. 1/13
C. 1/4
D. 3/4

Solution

  1. Step 1: Identify total outcomes

    A standard deck has 52 cards → total outcomes = 52.

  2. Step 2: Identify favourable outcomes

    There are 13 hearts in the deck → favourable count = 13.

  3. Step 3: Apply formula

    P(Heart) = 13 / 52 = 1/4.

  4. Final Answer:

    1/4 → Option C.

  5. Quick Check:

    13 hearts out of 52 → dividing by 13 gives 1/4 ✅
Hint: Remember each suit has 13 cards → probability for any suit = 13/52 = 1/4.
Common Mistakes: Using 26 (red cards) instead of 13 (hearts) when asked for a specific suit.
4. A bag contains 3 red balls and 2 blue balls. One ball is drawn at random. What is the probability that the drawn ball is red?
medium
A. 2/5
B. 3/5
C. 1/2
D. 3/4

Solution

  1. Step 1: Identify total outcomes

    Total balls = 3 red + 2 blue = 5.

  2. Step 2: Identify favourable outcomes

    Favourable (red) = 3.

  3. Step 3: Apply formula

    P(red) = 3 / 5 = 3/5.

  4. Final Answer:

    3/5 → Option B.

  5. Quick Check:

    3 red out of 5 total → 3/5 = 0.6 ✅
Hint: Add counts for total, then place the favourable count over total.
Common Mistakes: Swapping counts (using blue count instead of red) or forgetting to sum total balls first.
5. Two fair coins are tossed. What is the probability of getting at least one Head?
medium
A. 1/4
B. 1/2
C. 3/8
D. 3/4

Solution

  1. Step 1: Identify total outcomes

    For two coins the sample space is {HH, HT, TH, TT} → total outcomes = 4.

  2. Step 2: Identify favourable outcomes

    At least one Head = {HH, HT, TH} → favourable count = 3.

  3. Step 3: Apply formula

    P(at least one Head) = 3 / 4 = 3/4.

  4. Final Answer:

    3/4 → Option D.

  5. Quick Check:

    Only TT has no head → 1 outcome without head → 1/4 no-head, so at least one head = 1 - 1/4 = 3/4 ✅
Hint: Sometimes easier to find complement: P(at least one Head) = 1 - P(no Heads).
Common Mistakes: Counting HT and TH as the same outcome (they are distinct), leading to wrong totals.

Mock Test

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