Theoretical Probability
Theoretical probability is the probability calculated by analysing all possible outcomes of an experiment without actually performing it. It assumes that each outcome is equally likely.
Unlike experimental probability, which requires repeated trials, theoretical probability is determined purely by reasoning. For example, when tossing a fair coin, there are exactly 2 equally likely outcomes, so P(Head) = 1/2 — no experiment is needed.
In Class 10 NCERT, theoretical probability is the standard approach for solving problems involving coins, dice, cards, and drawing objects from bags.
What is Theoretical Probability?
Definition: Theoretical probability of an event E is the ratio of the number of outcomes favourable to E to the total number of equally likely outcomes in the sample space.
P(E) = Number of favourable outcomes / Total number of outcomes
Key terms:
- Experiment: An action that produces well-defined outcomes (e.g., rolling a die).
- Sample space (S): The set of all possible outcomes. For a die: S = {1, 2, 3, 4, 5, 6}.
- Event (E): A subset of the sample space. E.g., "getting an even number" = {2, 4, 6}.
- Favourable outcomes: Outcomes in the event set E.
- Equally likely outcomes: Each outcome has the same chance of occurring.
Conditions for theoretical probability to apply:
- All outcomes must be equally likely.
- The total number of outcomes must be finite and countable.
- The experiment must be well-defined (no ambiguity in outcomes).
Theoretical Probability Formula
Theoretical Probability Formula:
P(E) = n(E) / n(S)
Where:
- P(E) = probability of event E
- n(E) = number of outcomes favourable to E
- n(S) = total number of outcomes in the sample space
Complementary Event:
P(not E) = 1 − P(E)
Range of probability:
- P(E) is always between 0 and 1 inclusive: 0 ≤ P(E) ≤ 1.
- P(E) = 0 means the event is impossible.
- P(E) = 1 means the event is certain (sure event).
- P(E) + P(not E) = 1 always holds.
Derivation and Proof
Why does the formula work?
The theoretical probability formula follows from the assumption of equally likely outcomes:
- If every outcome in the sample space has the same chance, each outcome has probability 1/n(S).
- An event E consists of some outcomes from the sample space — say n(E) of them.
- Since each favourable outcome contributes 1/n(S), the total probability of E is: P(E) = n(E) × (1/n(S)) = n(E)/n(S).
- The complement (not E) contains n(S) − n(E) outcomes.
- So P(not E) = [n(S) − n(E)] / n(S) = 1 − n(E)/n(S) = 1 − P(E).
Theoretical vs Experimental Probability:
| Feature | Theoretical Probability | Experimental Probability |
|---|---|---|
| Based on | Reasoning and counting outcomes | Actual trials and observations |
| Requires experiment? | No | Yes |
| Result | Fixed for a given experiment | Varies with number of trials |
| Assumption | Equally likely outcomes | None |
| As trials → ∞ | Unchanged | Approaches theoretical probability |
Types and Properties
Types of events based on probability value:
- Sure event: P(E) = 1. Example: Getting a number less than 7 when rolling a standard die.
- Impossible event: P(E) = 0. Example: Getting 7 when rolling a standard die.
- Equally likely events: Events with the same probability. Example: Getting Head or Tail on a fair coin — both have P = 1/2.
- Complementary events: E and (not E) are complementary. P(E) + P(not E) = 1.
Common sample spaces:
| Experiment | Sample Space | n(S) |
|---|---|---|
| One coin toss | {H, T} | 2 |
| Two coin tosses | {HH, HT, TH, TT} | 4 |
| One die roll | {1, 2, 3, 4, 5, 6} | 6 |
| Two dice rolls | {(1,1), (1,2), ..., (6,6)} | 36 |
| One card from deck | 52 cards | 52 |
Methods
Steps to solve theoretical probability problems:
- Identify the experiment: What action is being performed? (tossing a coin, rolling a die, drawing a card, etc.)
- List the sample space: Write out all possible outcomes. Ensure they are equally likely.
- Count n(S): Total number of outcomes.
- Identify the event E: What specific outcome(s) are being asked about?
- Count n(E): Number of favourable outcomes.
- Apply the formula: P(E) = n(E) / n(S).
- Simplify: Reduce the fraction to lowest terms.
Common mistakes to avoid:
- Listing outcomes that are NOT equally likely. Example: When tossing 2 coins, listing {0 heads, 1 head, 2 heads} is wrong — these three are not equally likely. The correct sample space is {HH, HT, TH, TT}.
- Confusing "at least one" with "exactly one".
- Forgetting that a standard deck has 52 cards (not 54 — jokers are excluded).
- Not counting all outcomes for two dice — there are 36 pairs, not 12 or 21.
Solved Examples
Example 1: Probability of an Event on a Die
Problem: A fair die is rolled once. Find the probability of getting a number greater than 4.
Solution:
Given:
- Sample space S = {1, 2, 3, 4, 5, 6}, so n(S) = 6
- Event E = getting a number greater than 4 = {5, 6}
Using P(E) = n(E)/n(S):
- n(E) = 2
- P(E) = 2/6 = 1/3
Answer: P(number > 4) = 1/3
Example 2: Probability of a Prime Number on a Die
Problem: Find the probability of getting a prime number when a die is rolled.
Solution:
Given:
- S = {1, 2, 3, 4, 5, 6}, n(S) = 6
- Prime numbers in S: {2, 3, 5}
Using the formula:
- n(E) = 3
- P(prime) = 3/6 = 1/2
Answer: P(prime number) = 1/2
Example 3: Drawing a Ball from a Bag
Problem: A bag contains 5 red, 3 blue, and 2 green balls. One ball is drawn at random. Find the probability that it is blue.
Solution:
Given:
- Total balls = 5 + 3 + 2 = 10, so n(S) = 10
- Blue balls = 3
Using the formula:
- P(blue) = 3/10
Answer: P(blue ball) = 3/10
Example 4: Probability of NOT Getting an Event
Problem: A card is drawn from a well-shuffled deck. Find the probability that it is NOT a face card.
Solution:
Given:
- Total cards = 52, so n(S) = 52
- Face cards (Jack, Queen, King) = 4 × 3 = 12
Using complementary probability:
- P(face card) = 12/52 = 3/13
- P(not a face card) = 1 − 3/13 = 10/13
Answer: P(not a face card) = 10/13
Example 5: Two Dice Rolled Together
Problem: Two dice are rolled simultaneously. Find the probability that the sum of numbers is 7.
Solution:
Given:
- Total outcomes when two dice are rolled = 6 × 6 = 36
- Pairs with sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
Using the formula:
- n(E) = 6
- P(sum = 7) = 6/36 = 1/6
Answer: P(sum is 7) = 1/6
Example 6: Probability from a Word
Problem: A letter is chosen at random from the word "MATHEMATICS". Find the probability of choosing a vowel.
Solution:
Given:
- Letters in MATHEMATICS: M, A, T, H, E, M, A, T, I, C, S → 11 letters total
- Vowels: A, E, A, I → 4 vowels
Using the formula:
- P(vowel) = 4/11
Answer: P(vowel) = 4/11
Example 7: Numbered Tickets from a Box
Problem: Tickets numbered 1 to 20 are mixed in a box. One ticket is drawn at random. Find the probability that the number on the ticket is a multiple of 3 or 5.
Solution:
Given:
- n(S) = 20
- Multiples of 3 from 1 to 20: {3, 6, 9, 12, 15, 18} → 6 numbers
- Multiples of 5 from 1 to 20: {5, 10, 15, 20} → 4 numbers
- Common (multiples of both 3 and 5): {15} → 1 number
Using inclusion-exclusion:
- n(E) = 6 + 4 − 1 = 9
- P(multiple of 3 or 5) = 9/20
Answer: P(multiple of 3 or 5) = 9/20
Example 8: Probability of a Leap Year Having 53 Sundays
Problem: Find the probability that a leap year has 53 Sundays.
Solution:
Given:
- A leap year has 366 days = 52 complete weeks + 2 extra days.
- The 52 weeks already contain 52 Sundays.
- For 53 Sundays, one of the 2 extra days must be a Sunday.
Possible pairs for the 2 extra days:
- {Sun-Mon, Mon-Tue, Tue-Wed, Wed-Thu, Thu-Fri, Fri-Sat, Sat-Sun}
- Total pairs = 7
- Favourable (containing Sunday): Sun-Mon and Sat-Sun → 2
Answer: P(53 Sundays in a leap year) = 2/7
Example 9: Sure and Impossible Events
Problem: A die is rolled. Find: (a) P(getting a number less than 7), (b) P(getting 8).
Solution:
(a) Event: number less than 7
- E = {1, 2, 3, 4, 5, 6} → n(E) = 6
- P(E) = 6/6 = 1
- This is a sure event.
(b) Event: getting 8
- E = {} (empty set) → n(E) = 0
- P(E) = 0/6 = 0
- This is an impossible event.
Answer: (a) P = 1, (b) P = 0
Example 10: Cards: Probability of Red King
Problem: One card is drawn from a standard deck. Find the probability of getting a red king.
Solution:
Given:
- Total cards = 52
- Kings = 4 (one per suit)
- Red kings = King of Hearts + King of Diamonds = 2
Using the formula:
- P(red king) = 2/52 = 1/26
Answer: P(red king) = 1/26
Real-World Applications
Games and Sports:
- Calculating winning odds in board games, card games, and lotteries.
- Fair game design — ensuring no player has an unfair advantage.
Weather Forecasting:
- Probability models use theoretical frameworks to predict weather patterns.
Quality Control:
- Manufacturing uses probability to estimate defect rates in production batches.
Insurance:
- Insurance companies calculate premiums based on the theoretical probability of events (accidents, illness, natural disasters).
Genetics:
- Mendel's laws of inheritance use theoretical probability to predict offspring traits.
Key Points to Remember
- Theoretical probability is based on reasoning, not experiments: P(E) = n(E)/n(S).
- It applies only when all outcomes are equally likely.
- The probability of any event lies between 0 and 1: 0 ≤ P(E) ≤ 1.
- P(sure event) = 1 and P(impossible event) = 0.
- For complementary events: P(E) + P(not E) = 1.
- When tossing n coins, the total outcomes = 2ⁿ.
- When rolling n dice, the total outcomes = 6ⁿ.
- A standard deck has 52 cards: 4 suits × 13 cards each.
- As the number of experimental trials increases, experimental probability approaches theoretical probability.
- Always list the complete sample space before counting favourable outcomes.
Practice Problems
- A die is thrown once. Find the probability of getting an odd number.
- A bag contains 4 white, 6 black, and 5 red balls. One ball is drawn at random. Find the probability that it is neither white nor red.
- Two dice are thrown simultaneously. Find the probability that the product of numbers on the two dice is 12.
- A card is drawn from a well-shuffled deck. Find the probability that it is a black queen.
- Tickets numbered 1 to 50 are in a box. One ticket is drawn at random. Find the probability that the number is a perfect square.
- A letter is chosen at random from the word 'PROBABILITY'. Find the probability that it is a consonant.
- Find the probability that a non-leap year has 53 Mondays.
- A bag has 8 balls numbered 1 to 8. Find the probability of drawing a ball with a number divisible by 2 or 3.
Frequently Asked Questions
Q1. What is theoretical probability?
Theoretical probability is the probability of an event calculated by counting all possible equally likely outcomes without performing the experiment. The formula is P(E) = number of favourable outcomes / total number of outcomes.
Q2. What is the difference between theoretical and experimental probability?
Theoretical probability is calculated by reasoning (e.g., P(Head) = 1/2 for a fair coin). Experimental probability is calculated by performing actual trials (e.g., tossing a coin 100 times and counting heads). As the number of trials increases, experimental probability approaches theoretical probability.
Q3. What does 'equally likely outcomes' mean?
Equally likely outcomes means every outcome has the same chance of occurring. For a fair die, each face (1 through 6) has an equal chance of 1/6. If the die is biased, the outcomes are NOT equally likely and theoretical probability cannot be directly applied.
Q4. Can probability be negative or greater than 1?
No. Probability always lies between 0 and 1 inclusive: 0 ≤ P(E) ≤ 1. If your calculation gives a value outside this range, there is an error.
Q5. What is the probability of a sure event?
The probability of a sure event is 1. A sure event always occurs. Example: P(getting a number ≤ 6 on a die) = 6/6 = 1.
Q6. How do you find P(not E)?
Use the complementary rule: P(not E) = 1 − P(E). If P(getting a 6 on a die) = 1/6, then P(not getting a 6) = 1 − 1/6 = 5/6.
Q7. What is a sample space?
The sample space is the set of all possible outcomes of an experiment. For one coin toss: {H, T}. For two coin tosses: {HH, HT, TH, TT}. For one die: {1, 2, 3, 4, 5, 6}.
Q8. Why is theoretical probability important in CBSE Class 10?
Chapter 15 (Probability) in NCERT Class 10 is entirely based on theoretical probability. Questions on dice, coins, cards, and number problems regularly appear in board exams, carrying 3-5 marks.
Q9. How many cards are in a standard deck?
A standard deck has 52 cards: 4 suits (Hearts, Diamonds, Clubs, Spades) with 13 cards each (Ace through King). There are 26 red cards (Hearts + Diamonds) and 26 black cards (Clubs + Spades). Jokers are excluded.
Q10. When does theoretical probability fail?
Theoretical probability fails when outcomes are not equally likely. Example: predicting whether it will rain tomorrow — the outcomes 'rain' and 'no rain' are not equally likely. In such cases, experimental or statistical probability is used.
Related Topics
- Probability - Theoretical Approach
- Probability of Simple Events
- Experimental Probability
- Complementary Events in Probability
- Probability - Experimental Approach
- Probability of an Event
- Probability of Compound Events
- Impossible and Sure Events
- Probability with Playing Cards
- Probability with Dice
- Probability with Coins
