Chapter 14: Probability - Board Exam Notes

Exam Weightage & Blueprint

Total: ~4-5 Marks

Probability is one of the easiest chapters to score full marks. The board focuses on sample spaces of Coins, Dice, and Playing Cards.

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.

Self-Assessment Mock Test (10 Marks)

Q1 (1M): If $P(E) = 0.05$, what is the probability of 'not E'?

Q2 (2M): A bag contains 3 red balls and 5 black balls. A ball is drawn at random. What is the probability that the ball drawn is (i) red? (ii) not red?

Q3 (3M): A box contains 90 discs numbered 1 to 90. If one disc is drawn, find the probability that it bears (i) a two-digit number, (ii) a perfect square number.

Q4 (4M): (Case Study) A game consists of tossing a one rupee coin 3 times. Hanif wins if all tosses give the same result (3 heads or 3 tails). Calculate the probability that Hanif will lose the game.