Class 10 Probability Notes

Exam Weightage & Blueprint

Total: ~4-5 Marks

This chapter falls under Unit VII: Statistics & Probability (11 marks total). As per the latest syllabus: Classical definition of probability. Simple problems on finding the probability of an event.

Question Type	Marks	Frequency	Focus Topic
MCQ	1	Very High	Basic Probability Formula, Dice/Cards
Short Answer	2 or 3	High	2 Dice problems, Card properties
Case Study	4	Medium	Real-life scenarios (Balls in bag, Spinning wheel)

📐 Important Definitions & Formulas

1. Theoretical Probability

The theoretical probability of an event E is defined as:

$$ P(E) = \frac{\text{Number of outcomes favourable to } E}{\text{Number of all possible outcomes}} $$

(Pierre Simon Laplace, 1795)

2. Complementary Events

The event representing "not E" is called the complement of E, denoted by $\overline{E}$.

$$ P(E) + P(\overline{E}) = 1 \quad \Rightarrow \quad P(\overline{E}) = 1 - P(E) $$

3. Range of Probability

Rule: The probability of an event lies between 0 and 1 (inclusive).
$$ 0 \le P(E) \le 1 $$

Sure Event: Probability is 1 (Certain to happen).
Impossible Event: Probability is 0 (Cannot happen).

🎯 Mastering Sample Spaces

Most errors happen because students count total outcomes incorrectly. Memorize these standard experiments.

1. 💰 Coins

Experiment	Total Outcomes	Sample Space
1 Coin	2	{H, T}
2 Coins	4	{HH, HT, TH, TT}
3 Coins	8	{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

[Image of coin probability tree diagram]

2. 🎲 Dice

1 Die: Outcomes are {1, 2, 3, 4, 5, 6}. Total = 6.
Prime Numbers: 2, 3, 5 (Note: 1 is NOT prime).

2 Dice: Total Outcomes = $6 \times 6 = 36$.
Important for sums (e.g., Sum=8: (2,6), (3,5), (4,4), (5,3), (6,2)).

3. 🃏 Playing Cards

Total Cards = 52. Divided into 4 Suits of 13 cards each.

Color	Suit	Cards
Red (26)	Hearts ?, Diamonds ?	A, 2-10, J, Q, K
Black (26)	Spades ?, Clubs ?	A, 2-10, J, Q, K

Face Cards: Kings, Queens, Jacks. Total = 12 (3 per suit). Aces are NOT face cards!

Solved Examples (Board Marking Scheme)

Q1. One card is drawn from a well-shuffled deck of 52 cards. Calculate the probability that the card will be (i) be an ace, (ii) not be an ace. (2 Marks)

Step 1: Identify Total Outcomes 0.5 Mark

Total number of cards = 52.

Step 2: Probability of Ace (E) 0.5 Mark

Number of Aces = 4.

$P(E) = \frac{4}{52} = \frac{1}{13}$.

Step 3: Probability of Not Ace (F) 1 Mark

$P(F) = 1 - P(E) = 1 - \frac{1}{13} = \frac{12}{13}$.

Alternatively: Non-ace cards = $52 - 4 = 48$. $P(F) = \frac{48}{52} = \frac{12}{13}$.

Q2. Two dice are thrown at the same time. What is the probability that the sum of the two numbers appearing on the top of the dice is 8? (2 Marks)

Step 1: Total Outcomes 0.5 Mark

Total outcomes when throwing 2 dice = $6 \times 6 = 36$.

Step 2: Favourable Outcomes 1 Mark

Pairs with sum 8: $(2,6), (3,5), (4,4), (5,3), (6,2)$.

Number of favourable outcomes = 5.

Step 3: Calculation 0.5 Mark

$P(\text{sum is 8}) = \frac{5}{36}$.

Exam Strategy & Mistake Bank

⚠️ Mistake Bank

Probability Range: Never write an answer greater than 1 (e.g., 1.5) or negative. It is an instant 0 marks.

Prime Numbers: In dice problems, students often count 1 as a prime number. Remember: Primes are 2, 3, 5.

"At least": "At least one Head" in 2 coins means {HT, TH, HH} (3 outcomes), not just 1.

💡 Scoring Tips

Simplify Fractions: Always simplify your final answer (e.g., write $\frac{1}{2}$ instead of $\frac{26}{52}$).

Show Sample Space: For 2-3 mark questions, explicitly write the favourable outcomes (e.g., "Favourable outcomes: {2, 3, 5}").

Sum Rule: The sum of probabilities of all elementary events of an experiment is 1. Use this to check your answers.

📝 More Solved Board Questions

Q3. A card is drawn at random from a well-shuffled deck of 52 cards. Find the probability of getting (i) a king of red color, (ii) a red face card. 3 Marks

Sol. Total Cards = 52.

(i) Kings of Red color = 2 (King of Hearts and King of Diamonds).

$P(\text{Red King}) = 2/52 = \mathbf{1/26}$.

(ii) Face cards per suit = 3. Suits of red color = 2.

Total red face cards = $3 \times 2 = 6$.

$P(\text{Red Face Card}) = 6/52 = \mathbf{3/26}$.

🎯 Board Pattern (2018–2025): Questions often specify color AND type (e.g., "Black Queen" or "Red Ace"). Always count carefully: 26 red, 26 black; 13 per suit; 4 per rank.

📋 Board Revision Checklist

✅ $P(E) = \text{Fav outcomes} / \text{Total outcomes}$
✅ Range: $0 \le P(E) \le 1$
✅ $P(E) + P(\text{not } E) = 1$
✅ Sure event = 1; Impossible event = 0
✅ 2 Coins: 4 outcomes; 3 Coins: 8 outcomes
✅ 2 Dice: 36 outcomes
✅ 52 Cards: 12 face cards, 4 suits, 26 red/black
✅ "At least" means that value or MORE
✅ "At most" means that value or LESS

💡 Exam Tip:
If you are unsure of the sample space for 2 dice, quickly draw the $6 \times 6$ table on your rough sheet. It takes 1 minute but guarantees accuracy for sum-related questions.

Concept Mastery Quiz 🎯

Test your readiness for the board exam.

1. Which of the following cannot be the probability of an event?

2. An event is very unlikely to happen. Its probability is closest to:

3. If a card is selected from a deck of 52 cards, the probability of its being a red face card is:

4. A die is thrown once. The probability of getting a prime number is:

5. A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball is double that of a red ball, number of blue balls is:

Chapter 14: Probability

Overview