Probability Formulas are essential mathematical tools used in calculating the probability. Below is the main formula for probability.
Probability FormulaProbability of an Event = (Count of favorable outcomes) / (Total number of possible outcomes for the event)
P(A) = n(E) / n(S)
0 ≤ 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
Learn,
Some of the most common terms related to probability formulas 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 in 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. An example of a 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 that come from a random experiment.
- Equally Likely Events: Equally Likely Events have 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 a head or a tail, but we cannot get both at the same time.
Probability of an Event
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:
The probability P(E) lies between 0 and 1.
- For an impossible event (E), P(E) = 0.
- For a certain event (E), P(E) = 1.
The sum of the probabilities of all possible outcomes in a random experiment is equal to 1.
Example : In a rolling die experiment
Possible Outcomes : { 1, 2, 3, 4, 5, 6 }
then , P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1
The different Probability Formulas are discussed below:
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, and n(S) denotes the total count of outcomes within the sample space.
P(A) = Number of Favorable Outcomes/Total Number of Possible Outcomes
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)
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∣B) P(B) = P(B∣A)P(A)
- P(A∩B) is the joint probability, meaning the probability that both events A and B occur.
- P(A∣B) is the conditional probability of A given that B has already occurred.
- P(B∣A) is the conditional probability of B given that A has already occurred.
- P(A) and P(B) are the probabilities of events A and B occurring individually.
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. Then:
P(A∩B)=0
Thus, the Addition Rule for mutually exclusive events becomes:
P(A∪B) = P(A) + P(B)
If A is an event, then the probability of not A is expressed by the complementary rule:
P(not A) = 1 – P(A) or P(A’) = 1 – P(A).
P(A) + P(A′) = 1.
- P(A) is the probability that event A occurs.
- P(A') is the probability that event A does not occur.
- Since an event either happens or it doesn’t, their probabilities must add up to 1.
Some probability formulas based on complementary rules are as follows:
- P(A∩A′) = 0
- P(A′∩B) = P(B)−P(A∩B)
- P(A∩B′) = P(A)−P(A∩B)
- P(A∪B) = P(A∩B′) + P(A′∩B) + P(A∩B)
- P(A∩B) + P(A′∩B′) = 1 (Not always true)
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 the formula:
P(B∣A) = P(A∩B)/P(A)
P (B/A): Probability of event B when event A has already occurred.
The 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
The Multiplication Rule is used to find the probability of two or more events occurring together (simultaneously or in sequence). The formula depends on whether the events are independent or dependent.
- 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
Two events A and B are disjoint (or mutually exclusive) if they cannot happen at the same time. This means their intersection is empty:
P(A∩B) = 0
Bayes' Theorem
Bayes' Theorem calculates the probability of event A given the occurrence of event B. The Bayes Theorem Formula is given as
P(A∣B)= P(B∣A) × P(A)/ P(B)
- P(A∣B) = Probability of A happening given that B has occurred (posterior probability).
- P(B∣A) = Probability of B happening given that A has occurred.
- P(A) = Probability of A happening (prior probability).
- P(B) = Probability of B happening (total probability of evidence).
When two events depend on each other, the probability of one event affects the probability of the other. The formula for dependent probability is:
P(B and A) = P(A) × P(B | A)
- P(A∩B) = Probability of both A and B occurring.
- P(A) = Probability of A occurring first.
- P(B∣A) = Probability of B occurring, given that A has already occurred.
Two events A and B are independent if the occurrence of one does not affect the probability of the other.
For independent events, the probability of both occurring is:
P(A and B) = P(A) × P(B)
The Binomial Probability Formula is given as
P(x) = {}^nC_k· p^x (1 − p)^{n−x}
P(x) = [\frac{n!}{x!(n−x)!}]· p^x (1 − p)^{n−x}
Where,
n = Total number of events
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
The Normal probability formula is given by:
P(x) = (1/√2π)e^{(-x^2/2)}
Learn Normal Distribution
The formula for the experimental probability is;
Probability P(x) = Number of times an event occurs / Total number of trials.
The Theoretical Probability Formula is,
P(x) = Number of Favorable outcomes/ Number of Possible outcomes.
The Standard Deviation Probability Formula is given as
P(x) = (1/σ\sqrt{2\Pi}) e^{-(x-μ)^2/2σ^2}
A random variable X will have a 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.
In Class 10, we have to study basic probability, such as the probability of tossing a coin, tossing 2 coins, tossing 3 coins, throwing a die, throwing two dies, probability of drawing a card from well 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:
Various Probability Formulas |
|---|
Experimental or Empirical Probability Formula | P(E) = Number of times an event occurs / Total number of trials. |
Classical or Theoretical Probability Formula | P(E) = Number of Favorable Outcomes/Total Number of Possible Outcomes |
Addition Probability Formula | P(A ∪ B) = P(A) + P(B) – P(A∩B) |
Joint Probability 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 Probability Formula | P(x) = (1/√2П) e(-x2/2) |
Standard Deviation Probability 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
Event A = drawing a card with a feminine face
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
P(A) = 4/52
P(A) = 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) = 0.01
P(Smoke) = 0.20
P(Fire|Smoke) = 0.80
P(Fire|Smoke) = {P(Fire)P(Smoke Fire)}/P(Smoke)
We can substitute these values:
P(Fire | Smoke)=( 0.01 × 0.80)/ 0.20
P(Fire | Smoke)=0.008/0.20
P(Fire | Smoke)= 0.04 = 4%.
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:
We are given a bag containing:
- 2 Green bulbs
- 4 Orange bulbs
- 6 White bulbs
- Total bulbs = 2 + 4 + 6 = 12
We need to find the probability of picking either a green or a white bulb.
E = picking either a green bulb or a white bulb
P(E) = (Number of green bulbs + Number of white bulbs) / Total number of bulbs
P(E) = (2+6)/12
P(E) = 8/12
P(E) = 2/3.
Question 1. 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?
Question 2. 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?
Question 3. Drawing one card from a thoroughly shuffled deck of 52 cards, determine the probability that the card will:
- Be a king.
- Not to be a king.
Question 4. 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?
Question 5. Determine the probability of rolling an odd number when a six-sided die is rolled.