Probability Revision Notes - Class 10 Mathematics

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📌 Key Points

Probability is a measure of likelihood of an event occurring, ranging from 0 to 1
Theoretical probability: P(E) = Number of favorable outcomes / Total number of possible outcomes
Experimental probability: P(E) = Number of times event occurred / Total number of trials
Sample space is the set of all possible outcomes of an experiment
An event is a subset of the sample space
Complementary events: P(E) + P(E') = 1, meaning P(E') = 1 - P(E)
For mutually exclusive events: P(A or B) = P(A) + P(B)
For independent events: P(A and B) = P(A) × P(B)
Impossible event has probability 0; certain event has probability 1
For dependent events, the probability of the second event depends on the first
Law of Large Numbers: Experimental probability approaches theoretical as trials increase
Sample space for coin toss = {H, T}; for die roll = {1, 2, 3, 4, 5, 6}
For two dice, total outcomes = 6 × 6 = 36
Conditional probability: P(A|B) = P(A and B) / P(B)
Equally likely outcomes mean each outcome has same chance of occurring

📘 Important Definitions

Probability

Numerical measure of likelihood that an event will occur, expressed as a number between 0 and 1

Sample Space

Set of all possible outcomes of an experiment, denoted as S

Event

Subset of sample space; a set of outcomes of interest

Favorable Outcomes

Outcomes that result in the occurrence of the event we are interested in

Theoretical Probability

Probability calculated based on equally likely outcomes without conducting experiment

Experimental Probability

Probability calculated based on actual results from experiments or observations

Complementary Events

Two events that together make up the entire sample space; P(E) + P(E') = 1

Mutually Exclusive Events

Events that cannot occur together; P(A and B) = 0

Independent Events

Events where occurrence of one does not affect probability of the other; P(A and B) = P(A) × P(B)

Dependent Events

Events where occurrence of one affects the probability of the other

🔢 Formulas & Laws

Theoretical Probability

P(E) = n(E) / n(S)

where n(E) = number of favorable outcomes, n(S) = total number of possible outcomes

Experimental Probability

P(E) = Number of times E occurred / Total number of trials

Based on actual experimental data

Complementary Events

P(E') = 1 - P(E)

P(E) + P(E') = 1

Mutually Exclusive Events

P(A or B) = P(A) + P(B)

Events cannot occur together

Independent Events

P(A and B) = P(A) × P(B)

Probability of A is not affected by B

Total Outcomes (Two Trials)

Total outcomes = n₁ × n₂

If first experiment has n₁ outcomes and second has n₂ outcomes

Conditional Probability

P(A|B) = P(A and B) / P(B)

Probability of A given that B has occurred

⚠️ Common Mistakes

✗ Wrong: Confusing theoretical and experimental probability

✓ Correct: Theoretical probability is based on equally likely outcomes; experimental is based on actual data from trials

✗ Wrong: Using addition rule for independent events

✓ Correct: Use multiplication rule for independent events: P(A and B) = P(A) × P(B)

✗ Wrong: Not identifying sample space correctly

✓ Correct: Always list all possible outcomes before calculating probability

✗ Wrong: Assuming past outcomes affect future probability

✓ Correct: Each trial is independent (for fair experiments); past results don't influence future trials

✗ Wrong: Probability greater than 1 or less than 0

✓ Correct: Probability must always be between 0 and 1, inclusive

✗ Wrong: Forgetting to simplify fractions

✓ Correct: Always reduce probability fractions to lowest terms

✗ Wrong: Treating dependent events as independent

✓ Correct: When drawing without replacement, events are dependent; second draw is affected by first

📝 Exam Focus

These questions are frequently asked in CBSE exams:

Calculate theoretical and experimental probability from given data

Find probability of complementary events

Solve problems on mutually exclusive and independent events

Calculate probabilities with multiple trials (coins, dice, cards)

Apply probability to real-world scenarios (lottery, quality control, weather)

Distinguish between dependent and independent events

Find probability without replacement (drawing marbles, cards)

Use sample space and count favorable outcomes correctly

Solve conditional probability problems

🎯 Last-Minute Recall