MatHub

  
Probability
  
Introduction
  

    Probability is a branch of mathematics that deals with calculating the likelihood of events occurring. It quantifies uncertainty using values between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. Probability is widely used in statistics, science, finance, and everyday decision-making.
  

  
Probability
  

    
Definition: Probability measures the chance of an event happening. It is calculated as:
      $P\left(E\right)=\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$
    
    
Range: Probability values always lie between 0 and 1 (inclusive).
  

  
Key Terms Related to Probability
  

    
Experiment: A process with uncertain outcomes (e.g., tossing a coin).
    
Outcome: A single result of an experiment (e.g., getting heads).
    
Sample Space (S): The set of all possible outcomes of an experiment.
    
Event (E): A subset of the sample space (e.g., getting an even number when rolling a die).
    
Favorable Outcomes: Outcomes that satisfy the event’s condition.
    
Probability Line: A visual representation of probabilities from 0 to 1.
  

  
Some Experiments and Their Outcomes
  
1. Tossing a Coin
  

    
Sample Space: $\{\text{Heads},\text{Tails}\}$
    
Possible Events: Getting heads, getting tails.
  

  
2. Rolling a Die
  

    
Sample Space: $\{1,2,3,4,5,6\}$
    
Possible Events: Getting an odd number, getting a prime number.
  

  
3. Drawing a Card from a Deck
  

    
Sample Space: 52 cards (13 ranks × 4 suits).
    
4 suits are - Spades, Hearts , Clubs & Diamonds

    
13 cards in each suit
    
4 Aces
    
4 Kings
    
4 Queens
    
4 Jacks
    
Possible Events: Drawing a heart, drawing a king.
  

  
Probability of Occurrence of an Event
  
The probability of an event $E$ is calculated as:
  $P\left(E\right)=\frac{\text{Number of outcomes favorable to}E}{\text{Total number of outcomes in the sample space}}$

  
Examples:
  
Example 1: Tossing a Fair Coin
  
Problem: What is the probability of getting heads?
  
Solution:
  
Favorable outcomes = 1 (heads)
  
Total outcomes = 2 (heads, tails)
  
$P\left(\text{Heads}\right)=\frac{1}{2}$

  
Example 2: Rolling a Die
  
Problem: What is the probability of getting a number greater than 4?
  
Solution:
  
Favorable outcomes = 2 (5, 6)
  
Total outcomes = 6
  
$P\left(\text{Number > 4}\right)=\frac{2}{6}=\frac{1}{3}$

  
Example 3: Drawing a Marble from a Bag
  
Problem: A bag contains 3 red, 2 blue, and 5 green marbles. What is the probability of drawing a red marble?
  
Solution:
  
Favorable outcomes = 3 (red marbles)
  
Total outcomes = 3 + 2 + 5 = 10
  
$P\left(\text{Red}\right)=\frac{3}{10}$

 

Example: Drawing a King from a Deck of Cards
Problem: What is the probability of drawing a king from a deck of cards?
Solution:
Favorable outcomes = 4 (one king in each suit: hearts, diamonds, clubs, spades)
Total outcomes = 52 (total cards in a deck)
$P\left(\text{King}\right)=\frac{4}{52}=\frac{1}{13}$

Example: Drawing a Heart from a Deck of Cards
Problem: What is the probability of drawing a heart from a deck of cards?
Solution:
Favorable outcomes = 13 (one heart in each rank)
Total outcomes = 52 (total cards in a deck)
$P\left(\text{Heart}\right)=\frac{13}{52}=\frac{1}{4}$


  
Special Cases of Probability
  

    
Impossible Event: Probability = 0 (e.g., rolling a 7 on a standard die).
    
Certain Event: Probability = 1 (e.g., the sun rising tomorrow).
  

  
Applications of Probability
  

    
Statistics: Used in hypothesis testing and data analysis.
    
Weather Forecasting: Predicting rain, storms, or sunshine.
    
Finance: Assessing risks in investments and insurance.
    
Games: Designing fair rules for board games, card games, and lotteries.