Table of Contents
Probability measures the likelihood that a specific event will occur. It plays a crucial role in our daily lives, where we often need to predict the outcome of various events. Sometimes, we are confident about an outcome while some other times, we are uncertain about the outcomes. In such scenarios, probability helps us express this uncertainty by quantifying the chances of an event happening or not happening.
Probability is not just a mathematical concept; it is a powerful tool that helps us make decisions under uncertainty. Probability is widely used in various fields, including games, business forecasting, and even in advanced fields like artificial intelligence, where it helps in making informed predictions.
How to Calculate Probability
The probability of an event can be determined using a simple formula:
Probability = Number of Favorable Outcomes / Total Number of Possible Outcomes
- The probability value of any event lies between 0 and 1.
- A probability of 0 means the event cannot happen.
- A probability of 1 means a certainty that the event will occur.
- The favourable outcomes must always be non-negative and cannot exceed the total number of possible outcomes.
What is Probability?
Probability is a concept that measures the chance or likelihood of a specific event occurring. It is defined as the ratio of the number of favourable outcomes to the total number of possible outcomes. For an experiment with ‘n’ total outcomes, if ‘x’ represents the number of favourable outcomes, the probability of the event can be calculated using the following formula:
Probability = Number of Favorable Outcomes / Total Number of Possible Outcomes = x/n
Probability is frequently used to predict outcomes in various situations, such as tossing a coin, rolling a dice, or drawing a card from a deck of playing cards.
Types of Probability
Probability can be broadly classified into two main types:
- Theoretical Probability: This type of probability is based on the possible outcomes in a perfect world without any experimentation. It relies purely on the nature of the event.
- Experimental Probability: This type is determined by conducting an experiment and observing the actual outcomes. It depends on empirical data gathered from real-life trials.
Terminology in Probability Theory
Below given are some key terms that provide a better understanding of the basics of probability:
- Experiment: An experiment refers to a trial or operation performed to observe or produce an outcome. For example, tossing a coin or rolling a die are common experiments in probability.
- Sample Space: The sample space is the set of all possible outcomes of an experiment. For example, when tossing a coin, the sample space is {Head, Tail}.
- Favourable Outcome: A favourable outcome is an event that produces the desired result. For example, when rolling two dice, the favourable outcomes of getting a sum of 4 are (1, 3), (2, 2), and (3, 1).
- Trial: A trial refers to the act of performing a random experiment once. Each time you toss a coin or roll a die, you are conducting a trial.
- Random Experiment: A random experiment is an experiment that produces uncertain outcomes, but the set of possible outcomes is well-defined. For instance, when tossing a coin, you know the outcome will be either a head or a tail, but the result of each toss is uncertain.
- Event: An event is any specific set of outcomes from a random experiment. For example, rolling a die and getting an odd number (1, 3, or 5) is an event.
- Equally Likely Events: These are events that have the same probability of occurring. The outcome of one event does not affect the outcome of another. For instance, when tossing a coin, the chances of getting heads or tails are equal.
- Exhaustive Events: Exhaustive events are a set of outcomes that cover all possible results of an experiment, equivalent to the entire sample space.
- Mutually Exclusive Events: Mutually exclusive events are events that cannot happen simultaneously. For example, a day cannot be both sunny and rainy at the same time; these events exclude each other.
Events in Probability
In probability theory, an event is defined as a set of outcomes from an experiment or a subset of the sample space. If P(E) represents the probability of an event E, then the following conditions apply:
- P(E)=0 if and only if E is an impossible event, meaning it cannot occur.
- P(E)=1 if and only if E is a certain event, meaning it is guaranteed to occur.
- Probability always lies within the range 0≤P(E)≤1.
Comparing Probabilities of Events
Given two events, “A” and “B”, we can compare their probabilities:
If P(A)>P(B), then event “A” is more likely to occur than event “B”.
Sample Space and Probability Formula
Sample Space (S): Sample Space (S) is the set of all possible outcomes of an experiment.
Number of Outcomes in the Sample Space (n(S)): The number of Outcomes in the Sample Space (n(S)) is the total count of possible outcomes.
The probability of an event E occurring can be calculated using the formula:
P(E)= n(E) / n(S)
Where:
- n(E) is the number of favourable outcomes for event E.
- n(S) is the total number of outcomes in the sample space.
Probability of an Event Not Occurring
The probability of an event E not occurring, denoted by E’, is given by:
P(E)= n(S) – n(E)/n(S) = 1 – n(E)/n(S)
This formula shows that the probability of an event not happening is simply 1 minus the probability of it happening.
Probability Formula
The probability formula helps determine the likelihood of an event occurring. It is defined as the ratio of the number of favourable outcomes to the total number of possible outcomes. The formula is given by:
P(A)= n(A) / n(S)
Where:
- P(A) is the probability of event A.
- n(A) is the number of favourable outcomes for event A.
- n(S) is the total number of possible outcomes in the sample space.
Different Probability Formulas
Addition Rule: This rule applies when calculating the probability of the union of two events, A and B.
P(A or B) = P(A) + P(B) − P(A∩B)
or equivalently,
P(A∪B) = P(A) + P(B) − P(A∩B)
Complementary Rule: This rule is used when an event complements another event. If A is an event, then the probability of the event not occurring is:
P(A′) = 1 − P(A)
It follows that:
P(A) + P(A′) = 1
Conditional Probability: This formula is used when the probability of one event depends on the occurrence of another event. If event A has already occurred, the conditional probability of event B is:
P(B∣A) = P(A∩B) / P(A)
Multiplication Rule: This rule applies when calculating the probability of the intersection of two events, A and B, coinciding.
For independent events (where one event does not affect the other):
P(A∩B)=P(A)⋅P(B)
For dependent events (where the occurrence of one event affects the other):
P(A∩B)=P(A)⋅P(B∣A)
Examples of Probability Calculation
Example 1: Find the probability of getting a number less than 5 when a die is rolled.
Solution: Sample Space: When rolling a die, the possible outcomes are S={1,2,3,4,5,6}. So, n(S)=6.
Favourable Outcomes: The numbers less than 5 are A={1,2,3,4}. So, n(A)=4.
Probability Calculation:
P(E)= n(E) / n(S)
Example 2: Find the probability of getting a sum of 9 when two dice are thrown.
Solution: Sample Space: When throwing two dice, there are a total of 6 × 6 = 36 possible outcomes.
Favourable Outcomes: To get a sum of 9, the favourable pairs are (3,6), (6,3), (4,5), (5,4). So, n(A)=4.
Probability Calculation: P(E)= n(E) / n(S)
Answer: The probability of getting a sum of 9 is P(E)= 4/36 = 1/9.
FAQs on Probability
What is the basic formula for calculating probability?
The probability of an event A is calculated using the formula: Probability = Number of Favorable Outcomes/Total Number of Possible Outcomes
How do you calculate the probability of independent events occurring together?
For two independent events A and B, the probability of both events occurring together is given by: P(A∩B)=P(A) P(B)
What is the difference between classical and empirical probability?
Classical Probability is based on theoretical analysis, assuming all outcomes are equally likely (e.g., flipping a fair coin). Empirical Probability is based on actual experiments or observations, using the frequency of events to estimate probabilities (e.g., rolling a biased die multiple times and recording results).
