Probability - Class 10 Notes
Probability is a branch of mathematics that deals with the likelihood of events occurring. It is a crucial topic in mathematics, statistics, and real-world applications. In Class 10, students are introduced to the concept of probability and its various applications. Let's explore probability in detail with examples:
1. Introduction to Probability:
Probability (P) is a number between 0 and 1, where 0 represents an impossible event, and 1 represents a certain event. The formula to calculate the probability of an event E is: P(E) = (Number of favorable outcomes) / (Total number of possible outcomes).
2. Basic Probability Concepts:
a) Sample Space:
The sample space (S) of an experiment is the set of all possible outcomes.
Example: When rolling a fair six-sided die, the sample space S = {1, 2, 3, 4, 5, 6}.
b) Event:
An event is a subset of the sample space that represents specific outcomes of interest.
Example: Getting an even number when rolling a die is an event: E = {2, 4, 6}.
3. Types of Events:
a) Simple Event:
A simple event is an event with a single outcome.
Example: Rolling a 3 on a fair six-sided die is a simple event.
b) Compound Event:
A compound event is an event with more than one outcome.
Example: Drawing a card from a deck and getting either a red card or a face card is a compound event.
4. Probability of an Event:
a) Equally Likely Events:
If all outcomes in the sample space are equally likely, the probability of an event E is: P(E) = (Number of favorable outcomes) / (Total number of outcomes).
Example: The probability of rolling a 4 on a fair six-sided die is 1/6.
b) Not Equally Likely Events:
If all outcomes are not equally likely, the probability of an event E is: P(E) = (Number of favorable outcomes) / (Total number of outcomes).
Example: Drawing an ace from a standard deck of cards is 4/52 = 1/13.
5. Probability of Compound Events:
a) Independent Events:
Two events A and B are independent if the occurrence of one event does not affect the occurrence of the other. The probability of two independent events A and B occurring is: P(A ∩ B) = P(A) × P(B).
Example: Flipping a fair coin twice and getting heads both times: P(H, H) = P(H) × P(H) = (1/2) × (1/2) = 1/4.
b) Dependent Events:
Two events A and B are dependent if the occurrence of one event affects the occurrence of the other. The probability of two dependent events A and B occurring is: P(A ∩ B) = P(A) × P(B|A).
Example: Drawing two cards from a deck without replacement and getting two red cards: P(R, R) = P(R) × P(R|first card is red) = (26/52) × (25/51) = 25/102.
6. Addition Rule of Probability:
For any two events A and B, the probability of either event A or event B occurring is given by: P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
Example: Rolling a fair six-sided die and getting either an even number or a multiple of 3: P(E ∪ M) = P(E) + P(M) - P(E ∩ M) = (3/6) + (2/6) - (1/6) = 4/6 = 2/3.
7. Conditional Probability:
Conditional probability represents the probability of an event occurring given that another event has already occurred. The conditional probability of event A given event B is: P(A|B) = P(A ∩ B) / P(B).
Example: Drawing a king from a standard deck of cards, given that the card drawn is a heart: P(King|Heart) = P(King ∩ Heart) / P(Heart) = (1/52) / (13/52) = 1/13.
8. Bayes' Theorem:
Bayes' Theorem is used to calculate the probability of an event based on prior knowledge. P(A|B) = P(B|A) × P(A) / P(B).
Example: A medical test for a disease is 98% accurate. The probability of a person having the disease is 1%. What is the probability that a person has the disease given that the test is positive?
P(Disease|Positive Test) = P(Positive Test|Disease) × P(Disease) / P(Positive Test).
P(Disease|Positive Test) = 0.98 × 0.01 / (0.98 × 0.01 + 0.02 × 0.99) ≈ 0.332.