What is the probability formula?

P(Event) = Number of favourable outcomes / Total number of outcomes. Probability always lies between 0 and 1.

What is sample space?

Sample space is the set of all possible outcomes of an experiment. For example, when a coin is tossed, sample space = {H, T}.

What is the probability of an impossible event?

The probability of an impossible event is 0. The probability of a certain event is 1. All probabilities lie between 0 and 1.

Class 10 Maths Chapter 5 Probability Solutions | Maharashtra Board SSC

This page provides complete solutions for Maharashtra Board Class 10 Maths Chapter 5 — Probability. Probability deals with the likelihood of events occurring. This chapter introduces classical probability, sample space, and solving problems involving coins, dice, cards, and coloured balls.

📋 Table of Contents

Introduction to Probability

Probability is the measure of the likelihood of an event occurring. It was formally developed by mathematicians Blaise Pascal and Pierre de Fermat. Probability helps us make predictions when outcomes are uncertain.

🔢 Probability Formulas

P(Event E) = n(E) / n(S) where n(E) = favourable outcomes, n(S) = total outcomes
0 ≤ P(E) ≤ 1 (probability always between 0 and 1)
P(E) = 0 → impossible event
P(E) = 1 → certain/sure event
P(E) + P(E') = 1 → P(E') = 1 − P(E) (complementary event)
Sample Space (S) = set of all possible outcomes
Event (E) = subset of sample space
Equally likely outcomes → each outcome has equal chance

Key Concepts and Terminology

Experiment: Any activity with uncertain outcomes (e.g., tossing a coin)
Sample Space (S): Complete list of all possible outcomes
Event (E): A specific outcome or set of outcomes we are interested in
Favourable outcomes: Outcomes that satisfy the condition of the event
Equally likely outcomes: Each outcome has the same probability

Standard Sample Spaces:

One coin: S = {H, T} → n(S) = 2
Two coins: S = {HH, HT, TH, TT} → n(S) = 4
Three coins: n(S) = 8
One die: S = {1,2,3,4,5,6} → n(S) = 6
Two dice: n(S) = 36
Pack of cards: n(S) = 52 (4 suits × 13 cards each)

Practice Set 5.1 — Basic Probability

Q: A bag contains 3 red and 5 blue balls. A ball is drawn at random. Find the probability of drawing (i) a red ball (ii) a blue ball.

Step 1: Total balls = 3 + 5 = 8. So n(S) = 8.

Step 2: (i) Favourable outcomes for red = 3. P(red) = 3/8

Step 3: (ii) Favourable outcomes for blue = 5. P(blue) = 5/8

Step 4: Check: P(red) + P(blue) = 3/8 + 5/8 = 8/8 = 1 ✓

✅ Answer: P(red) = 3/8, P(blue) = 5/8

Q: A box contains 5 red, 4 green, and 3 yellow marbles. One marble is picked randomly. Find probability of (i) red (ii) green (iii) yellow.

Step 1: Total = 5 + 4 + 3 = 12 marbles

Step 2: (i) P(red) = 5/12

Step 3: (ii) P(green) = 4/12 = 1/3

Step 4: (iii) P(yellow) = 3/12 = 1/4

Step 5: Sum = 5/12 + 4/12 + 3/12 = 12/12 = 1 ✓

✅ Answer: P(red)=5/12, P(green)=1/3, P(yellow)=1/4

Practice Set 5.2 — Coins and Dice

Q: Two coins are tossed simultaneously. Find probability of (i) exactly one head (ii) both heads (iii) no head

Step 1: Sample Space S = {{HH, HT, TH, TT}}. n(S) = 4

Step 2: (i) Exactly one head: {{HT, TH}} → P = 2/4 = 1/2

Step 3: (ii) Both heads: {{HH}} → P = 1/4

Step 4: (iii) No head (both tails): {{TT}} → P = 1/4

Step 5: Check: 1/2 + 1/4 + 1/4 = 1 ✓

✅ Answer: P(exactly one head) = 1/2, P(both heads) = 1/4, P(no head) = 1/4

Q: A die is thrown. Find probability of getting (i) a prime number (ii) a number divisible by 3 (iii) a number greater than 4

Step 1: Sample Space = {{1,2,3,4,5,6}}. n(S) = 6

Step 2: (i) Prime numbers: {{2,3,5}} → P = 3/6 = 1/2

Step 3: (ii) Divisible by 3: {{3,6}} → P = 2/6 = 1/3

Step 4: (iii) Greater than 4: {{5,6}} → P = 2/6 = 1/3

✅ Answer: P(prime) = 1/2, P(div by 3) = 1/3, P(>4) = 1/3

Q: Two dice are thrown simultaneously. Find P(sum = 7)

Step 1: Total outcomes n(S) = 6×6 = 36

Step 2: Favourable outcomes (sum=7): (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) → 6 outcomes

Step 3: P(sum = 7) = 6/36 = 1/6

✅ Answer: P(sum = 7) = 1/6

Q: Two dice are thrown. Find P(sum > 10)

Step 1: n(S) = 36

Step 2: Sum > 10 means sum = 11 or 12.

Step 3: Sum = 11: (5,6),(6,5) → 2 outcomes

Step 4: Sum = 12: (6,6) → 1 outcome

Step 5: Total favourable = 3

Step 6: P(sum > 10) = 3/36 = 1/12

✅ Answer: P(sum > 10) = 1/12

Practice Set 5.3 — Playing Cards

A standard deck has 52 cards = 4 suits (Hearts ♥, Diamonds ♦, Clubs ♣, Spades ♠) × 13 cards each (A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K). Red cards = Hearts + Diamonds = 26. Black cards = Clubs + Spades = 26. Face cards = J, Q, K = 12 total.

Q: A card is drawn from a well-shuffled pack of 52 cards. Find P(i) a king (ii) a red card (iii) a face card (iv) a diamond

Step 1: n(S) = 52

Step 2: (i) Kings: 4 (one per suit) → P = 4/52 = 1/13

Step 3: (ii) Red cards: 26 → P = 26/52 = 1/2

Step 4: (iii) Face cards (J,Q,K): 3×4 = 12 → P = 12/52 = 3/13

Step 5: (iv) Diamonds: 13 → P = 13/52 = 1/4

✅ Answer: P(king)=1/13, P(red)=1/2, P(face)=3/13, P(diamond)=1/4

Q: Find P(a card drawn is an ace or a king)

Step 1: n(S) = 52

Step 2: Aces = 4, Kings = 4

Step 3: Ace OR King = 4 + 4 = 8 cards

Step 4: P = 8/52 = 2/13

✅ Answer: P(ace or king) = 2/13

Practice Set 5.4 — Mixed Problems

Q: A number is chosen at random from 1 to 25. Find P(prime number)

Step 1: n(S) = 25

Step 2: Prime numbers from 1 to 25: 2,3,5,7,11,13,17,19,23 → 9 primes

Step 3: P(prime) = 9/25

✅ Answer: P(prime) = 9/25

Q: In a class of 40 students, 15 like Maths, 20 like Science, and 5 like both. A student is selected at random. Find P(likes Maths or Science).

Step 1: n(S) = 40

Step 2: Likes Maths or Science = 15 + 20 − 5 = 30 (using addition principle)

Step 3: P(Maths or Science) = 30/40 = 3/4

✅ Answer: P(Maths or Science) = 3/4

Complementary Events

The complementary event E’ (read as ‘E prime’ or ‘not E’) consists of all outcomes NOT in E. The fundamental rule is: P(E) + P(E’) = 1.

Q: The probability that it will rain tomorrow is 0.65. Find P(it will NOT rain).

Step 1: P(rain) = 0.65

Step 2: P(not rain) = 1 − P(rain) = 1 − 0.65 = 0.35

✅ Answer: P(no rain) = 0.35

Q: A bag has 8 balls: 3 red, 5 blue. P(not red) = ?

Step 1: P(red) = 3/8

Step 2: P(not red) = 1 − 3/8 = 5/8

Step 3: Alternatively: P(blue) = 5/8 = P(not red) ✓

✅ Answer: P(not red) = 5/8

Frequently Asked Questions

What is the probability of getting a head when a coin is tossed?

P(head) = 1/2. A fair coin has two equally likely outcomes — head and tail. Since only one of them is a head, probability = 1/2 or 0.5.

Can probability be greater than 1?

No. Probability is always between 0 and 1 (inclusive). P = 0 means impossible, P = 1 means certain. If your calculated probability is greater than 1, you have made an error.

How many marks does Chapter 5 carry in SSC board exam?

Probability carries 5 to 7 marks in the SSC Algebra paper. Typically there is one short problem (2 marks) and one long problem (3-4 marks). Problems on coins, dice, balls in bags, and playing cards are most common.

What is the difference between ‘or’ and ‘and’ in probability?

‘Or’ means at least one of the events happens (union). ‘And’ means both events happen simultaneously (intersection). For Class 10, focus on ‘or’ using the addition rule: P(A or B) = P(A) + P(B) − P(A and B).