Probability

Class 10 Maths - Chapter 14 | NCERT Formula Sheet

1. Basic Probability Formula

P(E) = Number of favorable outcomes / Total number of possible outcomes
P(E) = n(E) / n(S)

Where:
• P(E) = Probability of event E
• n(E) = Number of favorable outcomes
• n(S) = Total number of outcomes in sample space

2. Range of Probability

Certain Event

P(E) = 1
Event will definitely occur

Impossible Event

P(E) = 0
Event will never occur

Possible Event

0 = P(E) = 1
Any probability value between 0 and 1

3. Complementary Events

P(E) + P(not E) = 1
P(not E) = 1 - P(E)
P(E) = 1 - P(not E)
Note: 'not E' or E' is called complement of event E

4. Sum of Probabilities

For all elementary events of an experiment,
Sum of probabilities = 1
P(E1) + P(E2) + P(E3) + ... = 1

5. Probability of Sure and Impossible Events

Sure Event

P(sure event) = 1
Example: Getting a number = 6 when rolling a die

Impossible Event

P(impossible event) = 0
Example: Getting 7 when rolling a die

6. Sample Space for Common Experiments

Experiment Sample Space n(S)
Tossing 1 coin {H, T} 2
Tossing 2 coins {HH, HT, TH, TT} 4
Rolling 1 die {1, 2, 3, 4, 5, 6} 6
Rolling 2 dice 36 ordered pairs 36
Drawing 1 card from deck 52 cards 52

7. Important Probability Values

• Getting head in coin toss: P(H) = 1/2
• Getting tail in coin toss: P(T) = 1/2
• Getting even number on die: P(even) = 3/6 = 1/2
• Getting prime number on die: P(prime) = 3/6 = 1/2
• Getting number > 4 on die: P(>4) = 2/6 = 1/3
• Drawing a spade from deck: P(spade) = 13/52 = 1/4

8. Deck of Cards Probability

Types of Cards

• Total cards: 52
• Red cards: 26
• Black cards: 26
• Hearts: 13
• Diamonds: 13
• Clubs: 13
• Spades: 13

Face Cards

• Kings: 4
• Queens: 4
• Jacks: 4
• Total face cards: 12
• Number cards: 40 (Ace to 10)

9. Quick Reference Formulas

Formula Description
P(E) = n(E)/n(S) Basic probability formula
P(E') = 1 - P(E) Complement rule
0 = P(E) = 1 Range of probability
P(sure event) = 1 Probability of certain event
P(impossible) = 0 Probability of impossible event

10. Example Problems (NCERT Style)

Problem 1: A die is thrown once. Find probability of getting:
a) Prime number b) Number between 2 and 6 c) Odd number

Solution:
Sample space S = {1,2,3,4,5,6}, n(S) = 6
a) Prime numbers = {2,3,5}, n(E) = 3, P = 3/6 = 1/2
b) Numbers between 2 and 6 = {3,4,5}, n(E) = 3, P = 3/6 = 1/2
c) Odd numbers = {1,3,5}, n(E) = 3, P = 3/6 = 1/2
Problem 2: Two coins are tossed simultaneously. Find probability of getting:
a) At least one head b) At most one head c) No head

Solution:
Sample space S = {HH, HT, TH, TT}, n(S) = 4
a) At least one head = {HH, HT, TH}, n(E) = 3, P = 3/4
b) At most one head = {HT, TH, TT}, n(E) = 3, P = 3/4
c) No head = {TT}, n(E) = 1, P = 1/4
Problem 3: One card is drawn from well-shuffled deck of 52 cards. Find probability of getting:
a) King of hearts b) A face card c) A red queen

Solution:
n(S) = 52
a) King of hearts = 1 card, P = 1/52
b) Face cards = 12, P = 12/52 = 3/13
c) Red queens = 2 (hearts & diamonds), P = 2/52 = 1/26

11. Important Points to Remember

12. Special Cases

Equally Likely Events

When all outcomes have equal chance:
P(each outcome) = 1/n(S)

Mutually Exclusive

Events cannot occur together
Example: Getting head and tail in single coin toss

Exhaustive Events

Set of all possible outcomes
Sum of probabilities = 1

13. Common Mistake Prevention

  1. Always identify sample space correctly
  2. Count favorable outcomes carefully
  3. Use fractions in simplest form
  4. Check if 0 = P(E) = 1
  5. For complementary events, verify P(E) + P(E') = 1
  6. In coin/dice problems, list all outcomes systematically