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Basic Probability Definition

Introduction

Probability என்பது ஒரு event நிகழும் சாத்தியத்தை அளவிடுகிறது. இது conditional probability மற்றும் Bayes’ theorem போன்ற உயர் நிலை probability concepts-ஐ புரிந்து கொள்ளும் அடிப்படையாக உள்ளது. இந்த அடிப்படை வரையறையை கற்றுக்கொண்டால், coin toss, dice throw, card draw போன்ற எளிய events-இன் probability-ஐ எளிதாக கணக்கிடலாம்.

Pattern: Basic Probability Definition

Pattern

ஒரு event-இன் probability என்பது, favourable outcomes எண்ணிக்கையும் total possible outcomes எண்ணிக்கையும் உள்ள விகிதமாகும்.

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

Probability மதிப்புகள் எப்போதும் 0 மற்றும் 1 இடையில் இருக்கும்.

Step-by-Step Example

Question

ஒரு coin ஒருமுறை toss செய்யப்படுகிறது. (i) Head வருவதற்கான probability, (ii) Tail வருவதற்கான probability-ஐ கண்டறியவும்.

Solution

  1. Step 1: எல்லா possible outcomes-ஐ அடையாளம் காண்க

    ஒரு coin toss செய்யப்படும்போது, 2 possible outcomes உள்ளன - Head (H) அல்லது Tail (T).

  2. Step 2: total outcomes-ஐ எண்ணவும்

    total outcomes எண்ணிக்கை = 2.

  3. Step 3: favourable outcomes-ஐ எண்ணவும்

    Head வருவதற்கான favourable outcomes எண்ணிக்கை = 1 (H மட்டும்).

  4. Step 4: probability-ஐ கணக்கிடவும்

    ஆகவே, P(Head) = 1/2 மற்றும் 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 ✅ (சரியானது, ஏனெனில் மொத்த probability எப்போதும் 1 ஆக இருக்க வேண்டும்).

Quick Variations

1. coin-க்கு பதிலாக die toss செய்வது (even number வருவதற்கான probability).

2. பல நிற balls உள்ள bag-இலிருந்து ஒரு ball எடுப்பது (red ball எடுப்பதற்கான probability).

3. multiple coins அல்லது dice - combined outcomes-க்கான probability.

Trick to Always Use

  • Step 1: total possible outcomes-ஐ தெளிவாக அடையாளம் காண்க.
  • Step 2: கேட்கப்பட்ட event-க்கு மட்டும் favourable outcomes-ஐ எண்ணவும்.
  • Step 3: formula-ஐ பயன்படுத்தவும்: P(E) = Favourable / Total.

Summary

Summary

Basic Probability Definition pattern-இல்:

  • Probability = Favourable / Total.
  • எல்லா probability மதிப்புகளும் 0 மற்றும் 1 இடையே இருக்கும்.
  • எல்லா possible outcomes-இன் probabilities-ன் கூட்டுத்தொகை = 1.
  • coin, dice, card போன்ற எளிய experiments-க்கு மிகவும் பயனுள்ளது.

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.

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