Is it correct to say: Treating (1, 2) and (2, 1) as the same outcome when two dice are thrown?

The dice are different, so these are two separate outcomes. That is why n(S) = 36, not 21.

Is it correct to say: Counting aces as face cards?

Face cards are only jack, queen and king — 12 cards. Aces are not face cards.

Is it correct to say: Writing the formula upside down as n(S)/n(A)?

P(A) = favourable outcomes ÷ total outcomes = n(A)/n(S). The answer can never exceed 1.

Is it correct to say: Counting common outcomes twice in 'A or B' events?

For 'a multiple of 3 or 5' from 1 to 30, numbers like 15 and 30 belong to both lists — count them once.

Is it correct to say: Repeating digits when the question says 'without repetition'?

Numbers such as 22 or 55 are not allowed. Read the condition before listing the sample space.

Is it correct to say: Leaving the answer unsimplified or as a count?

Give P(A) as a fraction in lowest terms (or decimal / percentage), not just n(A).

📄 Formula sheet (PDF)

Probability

Random experiments, outcomes, sample space, events and the probability of an event for coins, dice, cards and numbers — Maharashtra SSC Std X Algebra Ch 5

📐 14 formulas✏️ 4 examples🎯 6 practice🏫 Maharashtra State Board📚 Class 10✓ 2025–26 syllabus

💡

Board Exam Tips

→Write the answer in four steps: S (or n(S)), the event A, n(A), and then P(A) = n(A)/n(S). Each step carries marks, and 'complete the activity' questions are built on these blanks.
→For two dice or two coins, write the sample space as ordered pairs: (1, 2) and (2, 1) are different outcomes.
→Learn the structure of a pack of 52 cards: 4 suits of 13, 26 red and 26 black, 12 face cards (J, Q, K) and 4 aces.
→For 'not' events, count the outcomes of 'not A' directly, or find P(A) and use P(A') = 1 − P(A).
→Simplify the fraction. Any probability less than 0 or more than 1 means a counting mistake.

📐 Formulas(14)

Sample Space

S = {all possible outcomes}, n (S) = number of outcomes

Event

A \subseteq S

Probability of an Event★ Board fav

P (A) = \frac{n ( A )}{n ( S )}

Symbol	Meaning
$P (A)$	Probability of event A
$n (A)$	Number of outcomes favourable to A
$n (S)$	Total number of outcomes in the sample space

Range of Probability★ Board fav

0 \leq P (A) \leq 1

Sure (Certain) Event

P (S) = 1

Impossible Event

P (\emptyset) = 0

Equally Likely Outcomes

P (each outcome) = \frac{1}{n ( S )}

Sum of Probabilities of All Outcomes

\sum P (each outcome) = 1

Complement of an Event

P (A^{'}) = 1 - P (A)

Tossing Coins

n (S) = 2^{n}

Throwing Dice

n (S) = 6^{n}

Counting Tip: Sum on Two Dice

Ways to get sum s = 6 - ∣ s - 7∣, 2 \leq s \leq 12

Pack of Playing Cards

n (S) = 52 = 4 \times 13

Two-Digit Numbers from k Different Non-Zero Digits

n (S) = k (k - 1) (without repetition), n (S) = k^{2} (with repetition)

✏️ Solved Examples

1Solved Exampleeasy3 steps

A fair die is thrown once. Find the probability of getting a prime number.

Write the sample space.

S = {1, 2, 3, 4, 5, 6}, n (S) = 6

2Solved Exampleboard5 steps

Two dice are thrown together. Find the probability that (i) the sum of the numbers is 8, (ii) the sum is a multiple of 4.

Each die has 6 outcomes, so there are 6 × 6 ordered pairs.

n (S) = 36

3Solved Exampleboard4 steps

A card is drawn at random from a well-shuffled pack of 52 cards. Find the probability that the card is (i) a red face card, (ii) not a king.

Sample space.

n (S) = 52

4Solved ExampleHOTS3 steps

Two-digit numbers are formed using the digits 2, 3, 5, 7 without repeating a digit. One such number is chosen at random. Find the probability that it is (i) greater than 50, (ii) a prime number.

List the sample space: 4 choices for the tens digit and 3 for the units digit.

S = {23, 25, 27, 32, 35, 37, 52, 53, 57, 72, 73, 75}, n (S) = 4 \times 3 = 12

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Treating (1, 2) and (2, 1) as the same outcome when two dice are thrown
✓The dice are different, so these are two separate outcomes. That is why n(S) = 36, not 21.
2
Counting aces as face cards
✓Face cards are only jack, queen and king — 12 cards. Aces are not face cards.
3
Writing the formula upside down as n(S)/n(A)
✓P(A) = favourable outcomes ÷ total outcomes = n(A)/n(S). The answer can never exceed 1.
4
Counting common outcomes twice in 'A or B' events
✓For 'a multiple of 3 or 5' from 1 to 30, numbers like 15 and 30 belong to both lists — count them once.
5
Repeating digits when the question says 'without repetition'
✓Numbers such as 22 or 55 are not allowed. Read the condition before listing the sample space.
6
Leaving the answer unsimplified or as a count
✓Give P(A) as a fraction in lowest terms (or decimal / percentage), not just n(A).

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
Three coins are tossed together. Find the probability of getting exactly two heads.
Q2
A bag contains 4 red, 6 white and 5 black balls. One ball is drawn at random. Find the probability that it is not white.
Q3
Cards numbered 1 to 30 are put in a box and one card is drawn at random. Find the probability that the number is a multiple of 3 or of 5.
Q4
For an experiment, n(S) = 40 and P(A) = 3/8. Find n(A).
Q5
Two dice are thrown. Find the probability that both dice show the same number.
Q6
Which of these cannot be the probability of an event: 0.85, 3/2, 17%, 0, −0.2?

📝 Notes

Probability

Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). In Std X the whole chapter rests on one formula, $P(A) = \frac{n(A)}{n(S)}$ — the skill is in writing the sample space correctly.

Key words

Random experiment: an experiment whose result cannot be predicted, though all possible results are known (tossing a coin, throwing a die).
Outcome: one possible result.
Sample space (S): the set of all outcomes; $n(S)$ is how many there are.
Event: a subset of the sample space, usually described in words ("getting an even number").

Writing the sample space

Coins: each coin gives H or T, so n coins give $2^n$ outcomes.
Dice: two dice give 36 ordered pairs. Write them as a 6 × 6 grid of pairs to avoid missing any.
Cards: 52 cards — learn the suits, colours and face cards.
Numbers from digits: list them systematically by tens digit.