Probability Formulas are important mathematical tools used in calculating the probability. Before knowing the probability formulas, we need to understand the concept of Probability in short. The possibility of the occurrence of a random event is defined by probability. A probability is a chance of prediction. Its applications extend across various domains including gaming strategies, the creation of forecasts based on probability in business, and the evolving field of artificial intelligence.
In this article, we will learn the meaning and definition of the Probability Formula and how to use these formulas in calculating Probability. We also see various terms related to Probability and different formulas to easily solve mathematical problems.
Probability Formulas are used in determining the possibilities of an event by dividing the number of favourable outcomes by the total possible outcomes. By using this formula, we can estimate the probability associated with a specific occurrence.
Mathematically, we can write this formula as:
P(A) = Number of favourable outcomes / Total number of possible outcomes
Probability Formula calculates the ratio of favourable outcomes to the entire set of possible outcomes. The probability value lies within a range of 0 to 1, signifying that favourable outcomes cannot surpass the total outcomes, and the negative value of favourable outcomes is not possible.
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How to Calculate Probability?
Probability of an Event = (Count of favorable outcomes) / (Total number of possible outcomes for the event)
P(A) = n(E) / n(S)
P(A) < 1
Here, P(A) signifies the probability of an event A, where n(E) is the count of favorable outcomes, and n(S) is the total number of possible outcomes for the event.
When considering the complementary event, represented as P(A’), which denotes the non-occurrence of event A. then the formula will be:
P(A’) = 1- P(A)
P(A’), is the opposite of event A, indicating that either event P(A) occurs or its complement P(A’) occurs.
Therefore, now we can say; P(A) + P(A’) = 1
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Some of the most common terms related to probability formula are:
- Experiment: An Experiment is an action or procedure conducted to generate a particular outcome.
- Sample Space: The Sample Space includes the complete potential outcomes that come from an experiment. For example, when flipping a coin, the sample space includes {head, tail}.
- Favorable Outcome: A favorable Outcome is the result that aligns with the intended or expected conclusion. In the case of rolling two dice, examples of favorable outcomes resulting to a sum of 4 are (1,3), (2,2), and (3,1).
- Trial: A trial denotes the execution of a random experiment.
- Random Experiment: A Random Experiment is characterized by a well-defined set of possible outcomes. The example of random experiment is tossing a coin, where the result could be either heads or tails. That means the result would be uncertain.
- Event: An Event denotes the total outcomes come from a random experiment.
- Equally Likely Events: Equally Likely Events are those events which has identical probabilities of occurrence. The outcome of one event does not impact the outcome of another.
- Exhaustive Events: An Exhaustive Event occurs when the set of all possible outcomes covers the complete sample space.
- Mutually Exclusive Events: Mutually Exclusive Events are those that cannot occur simultaneously. For example, when we toss the coin, the result will be either head or tail but we cannot get both at the same time.
In Probability theory, an event represents a set of possible outcomes derived from an experiment. It often forms a subset of the overall sample space. If we represent the probability of an event E as P(E), the following principles apply:
When event E is impossible, then P(E) = 0.
When event E is certain, then P(E) = 1.
The probability P(E) lies between 0 and 1.
Consider two events, A and B. The probability of event A, denoted as P(A), which is greater than the probability of event B, P(B).
For a particular event E, probability formula will be:
P(E)= n(E)/ n(S)
Here, n(E) represents the number of outcomes favorable to event E.
n(S) denotes the total count of outcomes within the sample space.
The different Probability Formulas are discussed below:
Classical Probability Formula
P(A) = Number of Favorable Outcomes/Total Number of Possible Outcomes
Addition Rule Formula
When we deal with an event that is the union of two separate events, for example A and B, the probability of the union will be:
P(A or B) = P(A) + P(B) – P(A∩B)
P(A ∪ B) = P(A) + P(B) – P(A∩B)
Joint Probabilty Formula
It represents the common elements that constitute the distinct subsets of both events A and B. The formula can be expressed as:
P (A ∩ B) = P (A).P (B)
Addition Rule for Mutually Exclusive Events
If events A and B are mutually exclusive, that means they cannot happen at the same time, the probability of either event occurring is equal to the sum of their respective probabilities.
P(A or B)=P(A)+P(B)
Complementary Rule Formula
If A is an event, then the probability of not A is expressed by complementary rule:
P(not A) = 1 – P(A) or P(A’) = 1 – P(A).
P(A) + P(A′) = 1.
Some probability formulas based on them are as follows:
P(A.A’) = 0
P(A.B) + P (A’.B’) = 1
P(A’B) = P(B) – P(A.B)
P(A.B’) = P(A) – P(A.B)
P(A+B) = P(AB’) + P(A’B) + P(A.B)
Conditional Rule Formula
In the case, where the occurrence of event A is already known, the probability of event B is going to occur, referred to as conditional probability. It can be calculated using formula:
P(B∣A) = P(A∩B)/P(A)
P (B/A): Probability (conditional) of event B when event A has occurred.
P (A/B): Probability (conditional) of event A when event B has occurred.
Relative Frequency Formula
Relative Frequency formula is based on frequencies observed in real world data. This formula is given as
P(A) = Number of Times Event A Occurs/Total Number of Trials or Observations
Probability Formula with the Multiplication Rule
In situations where an event represents the simultaneous occurrence of two other events, denoted as events A and B, the probabilities of both events happening simultaneously can be calculated by using this formulas:
P(A ∩ B) = P(A)⋅P(B) (in case of independent events)
P(A∩B) = P(A)⋅P(B∣A) (in case of dependent events)
Disjoint Event
Disjoint events are events that never occur at the same time. These are also known as mutually exclusive events.
P(A∩B) = 0
Bayes’ Theorem
Bayes’ Theorem calculates the probability of event A given the occurrence of event B. Baye’s Theorem Formula is given as
P(A∣B)= P(B∣A)×P(A)/ P(B)
Learn, Bayes’ Theorem
Dependent Probability Formula
Dependent Probabilty are events that are affected by the occurrence of other events.The formula for the Dependent Probability is,
P(B and A) = P(A)×P(B | A)
Independent Probability Formula
Independent Probability are events that are not affected by the occurrence of other events. The formula for the Independent Probability is,
P(A and B) = P(A)×P(B)
Binominal Probability Formula
The Binomial Probability Formula is given as
P(x) = nCx · px (1 − p)n−x or P(r) = [n!/r!(n−r)!]· pr (1 − p)n−r
Where, n = Total number of events
r or x = Total number of successful events.
p = Success Probability in a single trial.
nCr = [n!/r!(n−r)]!
1 – p = Probability of failure.
Learn, Binomial Distribution
Normal Probabilty Formula
The Normal probability formula is given by:
P(x) = (1/√2П) e(-x^2/2)
Learn, Normal Distribution
Experimental Probability formula
The formula for the experimental probability is;
Probability P(x) = Number of times an event occurs / Total number of trials.
Theoretical Probabilty Formula
The Theoretical Probability Formula is,
P(x) = Number of Favourable outcomes/ Number of Possible outcomes.
Standard Deviation Probabilty Formula
The standard Deviation Probability Formula is given as
[Tex]P(x) = (1/σ\sqrt{2\Pi}) e^{-(x-μ)^2/2σ^2}[/Tex]
Bernoulli Probability Formula
A random variable X will have Bernoulli distribution with probability p, the formula is,
P(X = x) = px (1 – p)1−x, for x = 0, 1 and P(X = x) = 0 for other values of x
Here, 0 is failure and 1 is the success.
Learn, Bernoulli Distribution
In Class 10, we have to study basic probability such as probability of tossing a coin, tossing 2 coins, tossing 3 coins, throwing a die, throwing two dies, probability of drawing a card from well shuffled deck. All these questions can be solved with only one formula. The Probability Formula Class 10 is given as
P(E) = n(E)/n(s)
Where,
P(E) is Probability of an Event
n(E) is number of trials in which Event Occurred
n(S) is number of Sample Space
The various formula used in Probability Class 12 is tabulated below:
|
|
Experimental or Emperical Probability formula
| Number of times an event occurs / Total number of trials.
|
Classical or Theortical Probability Formula
| Number of Favorable Outcomes/Total Number of Possible Outcomes
|
Addition Probabilty Formula
| P(A ∪ B) = P(A) + P(B) – P(A∩B)
|
Joint Probabilty Formula
| P (A ∩ B) = P (A).P (B)
|
Addition Rule for Mutually Exclusive Events
| P(A or B)=P(A)+P(B)
|
Complementary Rule Formula
| P(not A) = 1 – P(A) or P(A’) = 1 – P(A).
P(A) + P(A′) = 1
|
Conditional Rule Formula
| P(B∣A) = P(A∩B)/P(A)
|
Relative Frequency Formula
| P(A)= Number of Times Event A Occurs/Total Number of Trials or Observations
|
Disjoint Event
| P(A∩B) = 0
|
Bayes’ Theorem
| P(A∣B)= P(B∣A)×P(A)/ P(B)
|
Dependent Probability Formula
| P(B and A) = P(A)×P(B | A)
|
Independent Probability Formula
| P(A and B) = P(A)×P(B)
|
Binominal Probability Formula
| P(x) = nCx · px (1 − p)n−x or P(r) = [n!/r!(n−r)!]· pr (1 − p)n−r
|
Normal Probabilty Formula
| P(x) = (1/√2П) e(-x2/2)
|
Standard Deviation Probabilty Formula
| P(x) = (1/σ√2П) e-(x-μ)^2/2σ^2
|
Bernoulli Probability Formula
| P(X = x) = px (1 – p)1-x, for x = 0, 1 and P(X = x) = 0 for other values of x.
|
Also, Check
Example 1: Select a card at random from a standard deck. What is the probability of drawing a card with a feminine face?
Solution:
In a standard deck containing 52 cards: Total possible outcomes = 52
The number of favorable events (considering only queens as feminine faces) = 4
Therefore, the probability P(A) is calculated using the formula:
P(A) = Number of Favorable Outcomes ÷ Total Number of Outcomes
= 4/52
= 1/13.
Example 2: If the Probability of event E, denoted as P(E)=0.35, what is the probability of the complement event ‘not E’?
Solution:
Given that P(E)=0.35, we can use the complementary probability formula:
P(E) + P(not E) = 1
Substituting the known value:
P(not E) = 1 – P(E)
P(not E) = 1 – 0.35
Hence, P(not E) = 0.65
Example 3: Dangerous fires are very rare around 1% but the smoke is fairly common around 20% due to barbecues. Find the dangerous fire when 80% of dangerous fires produce smoke.
Solution:
Probability of dangerous Fire when there is smoke by using Bayes theorem:
P(Fire|Smoke) = {P(Fire)P(Smoke Fire)}/P(Smoke)
P(Fire)=0.01(1%) and P(Smoke|Fire)= 0.80 (80%), we can substitute these values:
P(Fire | Smoke)=( 0.02×0.90)/ 0.30
(Fire | Smoke)=0.018/0.30
(Fire | Smoke)= 0.06 = 6%.
Example 4: Within a bag, there are 2 green bulbs, 4 orange bulbs, and 6 white bulbs. When a bulb is randomly chosen from the bag, what is the probability of picking either a green bulb or a white bulb?
Solution:
Total number of bulbs in the bag is 2 green + 4 orange + 6 white = 12 bulbs
Number of green bulbs = 2, and the number of white bulbs = 6
Probability = (Number of green bulbs + Number of white bulbs) / Total number of bulbs
Probability = (2+6)/12
Probability = 8/12
Probability = 2/3.
Q1. From a collection of marbles in a bag—8 red, 9 blue, and 6 green—two marbles are randomly picked without replacement. What is the probability that both marbles selected are blue?
Q2. In a drawer containing 6 black pens, 4 blue pens, and 7 red pens, a pen is drawn at random. What is the probability that the pen is either black or blue?
Q3. Drawing one card from a thoroughly shuffled deck of 52 cards, determine the probability that the card will:
Q4. According to a survey, 70% of individuals enjoy chocolate, and among those chocolate enthusiasts, 60% also have a liking for vanilla. What is the probability that an individual likes vanilla, given their fondness for chocolate?
Q5. Determine the probability of rolling an odd number when a six-sided die is rolled.
1. What is Meaning of Probability?
The possibility of occurrence of a random event is defined by probability. A probability is a chance of prediction.
2. What is the Meaning of Probability Formula?
Probability Formulas are used in determining the possibilities of an event by dividing the number of favorable outcomes by the total possible outcomes. The probability value lies within a range 0 to 1, signifying that favorable outcomes cannot surpass the total outcomes, and the negative value of favourable outcomes is not possible.
3. What is the Meaning of notation U and ∩ mean in Probability?
The symbol “U” in probability denotes a uniform distribution. On the other hand, the symbol “∩” signifies the intersection of sets. In simpler terms, the intersection of two sets is the most extensive set involve all elements shared by both sets.
4. What is the Conventional Formula to calculate the Probability?
The Probability of an Event = (Count of favourable outcomes) / (Total number of possible outcomes for the event)
P(A) = n(E) / n(S)
P(A) < 1
Here, P(A) signifies the Probability of an event A, where n(E) is the count of favorable outcomes, and n(S) is the total number of possible outcomes for the event.
5. What is Complementary Formula?
If A is an event, then the probability of not A is expressed by complementary rule:
P(not A) = 1 – P(A) or P(A’) = 1 – P(A).
P(A) + P(A′) = 1.
6. What is Disjoint Event?
Disjoint events are events that never occur at the same time. These are also known as mutually exclusive events.
P(A∩B) = 0.
7. What is Bayes’ Theorem?
P(A∣B)= P(B∣A)×P(A)/ P(B)
Bayes’ Theorem calculates the probability of event A given the occurrence of event B.
8. What is Conditional Formula?
In the case, where the occurrence of event A is already known, the probability of event B is going to occur, referred to as conditional probability. It can be calculated using formula:
P(B∣A) = P(A∩B)/P(A)
P (B/A): Probability (conditional) of event B when event A has occurred.
P (A/B): Probability (conditional) of event A when event B has occurred.
9. What are some real-life examples of Probability?
Weather Prediction, card games, political voting, dice games and flip a coin etc. are some examples of Probability
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