Probability - Theoretical Approach
Probability is the mathematical study of how likely an event is to occur. In CBSE Class 10 Mathematics, Chapter 15 (Probability) introduces the theoretical (classical) approach to probability.
In the experimental (empirical) approach studied in Class 9, probability was calculated by performing experiments and observing outcomes. In the theoretical approach, probability is calculated without performing the experiment -- by reasoning about the equally likely outcomes.
The theoretical approach assumes that all outcomes of a random experiment are equally likely. The probability of an event is the ratio of favourable outcomes to total outcomes.
What is Probability Class 10 - Theoretical Approach, Formula & Solved Examples?
Definition: The theoretical probability (or classical probability) of an event E is defined as:
P(E) = Number of favourable outcomes / Total number of outcomes
Key terms:
- Experiment -- an action or process that leads to well-defined outcomes (e.g., tossing a coin, rolling a die).
- Random experiment -- an experiment whose outcome cannot be predicted in advance.
- Outcome -- a possible result of the experiment.
- Sample space (S) -- the set of all possible outcomes.
- Event (E) -- a subset of the sample space; a collection of specific outcomes.
- Favourable outcomes -- outcomes that satisfy the condition of the event.
- Equally likely outcomes -- outcomes that have the same chance of occurring.
Properties of probability:
- 0 <= P(E) <= 1 for any event E.
- P(sure event) = 1 (an event that always happens).
- P(impossible event) = 0 (an event that never happens).
- P(E) + P(not E) = 1 (complementary events).
- P(not E) = 1 - P(E).
Probability - Theoretical Approach Formula
Probability Formula:
P(E) = Number of outcomes favourable to E / Total number of equally likely outcomes
Complementary Event:
P(not E) = 1 - P(E)
or equivalently: P(E) + P(E-bar) = 1
Common sample spaces:
| Experiment | Sample Space | Total Outcomes |
|---|---|---|
| Tossing 1 coin | {H, T} | 2 |
| Tossing 2 coins | {HH, HT, TH, TT} | 4 |
| Tossing 3 coins | 8 outcomes | 8 |
| Rolling 1 die | {1, 2, 3, 4, 5, 6} | 6 |
| Rolling 2 dice | 36 outcomes | 36 |
| Drawing 1 card from deck | 52 cards | 52 |
Derivation and Proof
Theoretical vs. Empirical Probability:
| Feature | Theoretical | Empirical |
|---|---|---|
| Basis | Reasoning about outcomes | Performing experiments |
| Formula | Favourable / Total | Frequency / Total trials |
| Requires experiment? | No | Yes |
| Assumption | Equally likely outcomes | Large number of trials |
| Accuracy | Exact (under assumptions) | Approximate (improves with more trials) |
Why does the formula work?
- If all outcomes are equally likely, each outcome has probability 1/n (where n = total outcomes).
- An event with m favourable outcomes has probability = m x (1/n) = m/n.
- This is the classical definition of probability, proposed by Laplace.
Connection to empirical probability:
As the number of trials increases, the empirical probability approaches the theoretical probability. This is known as the Law of Large Numbers.
Types and Properties
Problems on probability in Class 10 include:
Type 1: Single Coin/Die
- Find P(getting a head), P(getting an even number), P(getting a prime number), etc.
Type 2: Two Coins/Dice
- List the sample space and count favourable outcomes for events like "sum is 7" or "at least one head."
Type 3: Playing Cards
- Standard deck of 52 cards: find P(drawing a king), P(red card), P(face card), etc.
Type 4: Complementary Events
- Use P(not E) = 1 - P(E) to find probabilities of "not happening" events.
Type 5: Numbers and Divisibility
- A number is chosen at random from 1 to n. Find P(divisible by 3), P(perfect square), etc.
Type 6: Bags, Boxes, and Drawing
- Balls of different colours in a bag. Find P(drawing a specific colour).
Type 7: Word and Letter Problems
- A letter is chosen at random from a word. Find P(vowel), P(consonant), etc.
Methods
Step-by-step method for solving probability problems:
- Identify the experiment -- what action is being performed?
- List the sample space -- write out all possible outcomes.
- Count total outcomes (n).
- Identify the event -- what specific outcome or set of outcomes is being asked about?
- Count favourable outcomes (m).
- Apply the formula: P(E) = m/n.
For complementary events:
- P(at least one) = 1 - P(none).
- P(not E) = 1 - P(E).
Playing Cards Reference:
- Total: 52 cards (26 red, 26 black).
- 4 suits: Hearts (red), Diamonds (red), Clubs (black), Spades (black).
- Each suit: 13 cards (A, 2, 3, ..., 10, J, Q, K).
- Face cards: J, Q, K (12 total: 3 per suit).
- Aces: 4 (one per suit).
- Number cards: 2 to 10 (36 total: 9 per suit).
Common Mistakes:
- Not listing the complete sample space (missing outcomes).
- Confusing "at least one" with "exactly one."
- Assuming outcomes are equally likely when they are not.
- Getting probability greater than 1 or less than 0 (impossible).
Solved Examples
Example 1: Rolling a Die
Problem: A die is thrown once. Find the probability of getting: (a) a number greater than 4, (b) a prime number.
Solution:
Sample space: {1, 2, 3, 4, 5, 6}. Total outcomes = 6.
(a) Number greater than 4:
- Favourable outcomes: {5, 6} = 2
- P(greater than 4) = 2/6 = 1/3
(b) Prime number:
- Prime numbers in sample space: {2, 3, 5} = 3
- P(prime) = 3/6 = 1/2
Answer: (a) 1/3, (b) 1/2.
Example 2: Tossing Two Coins
Problem: Two coins are tossed simultaneously. Find the probability of getting: (a) exactly one head, (b) at least one head, (c) no head.
Solution:
Sample space: {HH, HT, TH, TT}. Total outcomes = 4.
(a) Exactly one head:
- Favourable: {HT, TH} = 2
- P = 2/4 = 1/2
(b) At least one head:
- Favourable: {HH, HT, TH} = 3
- P = 3/4 = 3/4
(c) No head:
- Favourable: {TT} = 1
- P = 1/4 = 1/4
Verification: P(at least one head) = 1 - P(no head) = 1 - 1/4 = 3/4. Confirmed.
Example 3: Drawing a Card from a Deck
Problem: A card is drawn at random from a well-shuffled deck of 52 cards. Find the probability of getting: (a) a king, (b) a red card, (c) a face card, (d) not a diamond.
Solution:
Total outcomes = 52.
(a) A king:
- Kings in deck = 4
- P = 4/52 = 1/13
(b) A red card:
- Red cards (Hearts + Diamonds) = 26
- P = 26/52 = 1/2
(c) A face card (J, Q, K):
- Face cards = 12 (3 per suit x 4 suits)
- P = 12/52 = 3/13
(d) Not a diamond:
- Diamonds = 13, so not diamond = 52 - 13 = 39
- P = 39/52 = 3/4
Example 4: Bag of Coloured Balls
Problem: A bag contains 5 red, 4 blue, and 3 green balls. A ball is drawn at random. Find the probability of getting: (a) a blue ball, (b) not a red ball, (c) a red or green ball.
Solution:
Total balls = 5 + 4 + 3 = 12.
(a) Blue ball:
- Favourable = 4
- P = 4/12 = 1/3
(b) Not a red ball:
- Not red = 4 + 3 = 7
- P = 7/12 = 7/12
Alternatively: P(not red) = 1 - P(red) = 1 - 5/12 = 7/12.
(c) Red or green:
- Favourable = 5 + 3 = 8
- P = 8/12 = 2/3
Example 5: Number Chosen at Random
Problem: A number is chosen at random from 1 to 20. Find the probability that: (a) it is divisible by 3, (b) it is a perfect square, (c) it is a prime number.
Solution:
Total outcomes = 20.
(a) Divisible by 3:
- Numbers: 3, 6, 9, 12, 15, 18 = 6 numbers
- P = 6/20 = 3/10
(b) Perfect square:
- Numbers: 1, 4, 9, 16 = 4 numbers
- P = 4/20 = 1/5
(c) Prime number:
- Primes: 2, 3, 5, 7, 11, 13, 17, 19 = 8 numbers
- P = 8/20 = 2/5
Example 6: Letters of a Word
Problem: A letter is chosen at random from the word MATHEMATICS. Find the probability of getting: (a) a vowel, (b) the letter M, (c) a consonant.
Solution:
Letters in MATHEMATICS: M, A, T, H, E, M, A, T, I, C, S = 11 letters.
(a) Vowel (A, E, I):
- Vowels: A, A, E, I = 4
- P = 4/11 = 4/11
(b) Letter M:
- M appears 2 times
- P = 2/11 = 2/11
(c) Consonant:
- Consonants = 11 - 4 = 7
- P = 7/11 = 7/11
Example 7: Two Dice Problem
Problem: Two dice are thrown simultaneously. Find the probability that the sum of numbers is: (a) 7, (b) more than 10, (c) less than 4.
Solution:
Total outcomes = 6 x 6 = 36.
(a) Sum = 7:
- Favourable: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6
- P = 6/36 = 1/6
(b) Sum > 10 (i.e., sum = 11 or 12):
- Sum 11: (5,6), (6,5) = 2
- Sum 12: (6,6) = 1
- Total favourable = 3
- P = 3/36 = 1/12
(c) Sum < 4 (i.e., sum = 2 or 3):
- Sum 2: (1,1) = 1
- Sum 3: (1,2), (2,1) = 2
- Total favourable = 3
- P = 3/36 = 1/12
Example 8: Complementary Event Problem
Problem: The probability of rain on a particular day is 0.3. What is the probability that it does NOT rain?
Solution:
Given: P(rain) = 0.3
Using complementary event:
- P(no rain) = 1 - P(rain) = 1 - 0.3 = 0.7
Answer: The probability of no rain is 0.7.
Example 9: Leap Year Birthday Problem
Problem: Find the probability that a leap year has 53 Sundays.
Solution:
A leap year has 366 days = 52 complete weeks + 2 extra days.
52 weeks give 52 Sundays. For 53 Sundays, one of the 2 extra days must be a Sunday.
The 2 extra days can be:
- (Sunday, Monday), (Monday, Tuesday), (Tuesday, Wednesday), (Wednesday, Thursday), (Thursday, Friday), (Friday, Saturday), (Saturday, Sunday)
- Total possibilities = 7
Favourable (at least one Sunday):
- (Sunday, Monday) and (Saturday, Sunday) = 2
P(53 Sundays) = 2/7 = 2/7.
Answer: The probability is 2/7.
Example 10: Card - Neither King nor Queen
Problem: A card is drawn from a deck of 52 cards. Find the probability that it is neither a king nor a queen.
Solution:
Total outcomes = 52.
- Kings = 4, Queens = 4
- Cards that are king or queen = 8
- Cards that are neither king nor queen = 52 - 8 = 44
P = 44/52 = 11/13.
Answer: P(neither king nor queen) = 11/13.
Real-World Applications
Weather Forecasting:
- Meteorologists use probability to predict the likelihood of rain, storms, or extreme weather events.
Insurance:
- Insurance companies use probability to calculate premiums based on the likelihood of claims for different age groups, health conditions, and risk profiles.
Medicine:
- The probability of a treatment being effective, the probability of side effects, and genetic probability of inheriting diseases.
Games and Sports:
- Probability governs card games, board games, and lotteries. Sports analytics uses probability to predict match outcomes.
Quality Control:
- Manufacturers calculate the probability of a product being defective to set quality standards.
Everyday Decisions:
- Probability reasoning helps in decisions like choosing routes (traffic probability), studying for exams (topic weightage), and risk assessment.
Key Points to Remember
- Theoretical probability: P(E) = Favourable outcomes / Total equally likely outcomes.
- 0 <= P(E) <= 1 for any event E.
- P(sure event) = 1, P(impossible event) = 0.
- P(E) + P(not E) = 1 (complementary events).
- Sample space for 1 coin = 2, 2 coins = 4, 3 coins = 8 outcomes.
- Sample space for 1 die = 6, 2 dice = 36 outcomes.
- Standard deck = 52 cards, 4 suits, 13 cards per suit, 12 face cards.
- "At least one" problems are best solved using complementary events: P(at least one) = 1 - P(none).
- Equally likely outcomes are essential for the classical formula to apply.
- As the number of trials increases, empirical probability approaches theoretical probability (Law of Large Numbers).
Practice Problems
- A die is thrown once. Find the probability of getting a number divisible by 2 or 3.
- Three coins are tossed simultaneously. Find the probability of getting exactly 2 heads.
- A card is drawn from a deck. Find the probability of getting a black face card.
- A bag has 3 red, 5 white, and 2 blue balls. Find the probability of NOT drawing a white ball.
- A number is chosen from 1 to 50. Find the probability that it is divisible by both 3 and 5.
- Find the probability of getting a sum of 9 when two dice are thrown.
- A letter is chosen at random from PROBABILITY. Find the probability of it being a vowel.
- The probability of a student passing is 0.85. Find the probability of the student failing.
Frequently Asked Questions
Q1. What is theoretical probability?
Theoretical probability is calculated by reasoning about equally likely outcomes, without performing an experiment. P(E) = Favourable outcomes / Total outcomes.
Q2. How is theoretical probability different from empirical probability?
Theoretical probability uses reasoning (no experiment needed). Empirical probability uses actual experiment results (frequency/total trials). As trials increase, empirical approaches theoretical.
Q3. Can probability be negative or greater than 1?
No. Probability always lies between 0 and 1 inclusive. P = 0 means impossible, P = 1 means certain.
Q4. What is a complementary event?
The complement of event E (written as E-bar or not-E) consists of all outcomes NOT in E. P(E) + P(not E) = 1.
Q5. How many outcomes when two dice are thrown?
36 outcomes (6 x 6). Each die has 6 faces, and each face of one die can pair with each face of the other.
Q6. How many face cards in a deck of 52 cards?
12 face cards: Jack, Queen, King in each of the 4 suits (3 x 4 = 12). Some questions also count Aces as face cards -- check the problem statement.
Q7. What does 'at least one' mean in probability?
'At least one' means one or more. The easiest way to calculate P(at least one) = 1 - P(none).
Q8. What is a sure event and an impossible event?
A sure event always happens (P = 1), e.g., getting a number less than 7 on a die. An impossible event never happens (P = 0), e.g., getting 7 on a standard die.
