104 LearnersLast updated on June 18th, 2025
Probability Distribution
The probability distribution is a statistical function that calculates the relative likelihood of all possible experimental outcomes. It tells us the way by which the values of a random variable are given out. Probability distribution helps in predicting weather conditions or test results .
What is a Probability Distribution?
A probability distribution defines all the potential values and probabilities for a random variable within a specific range. This range will be restricted to the lowest and highest possible values. Several factors influence where the potential value will be plotted on the probability distribution. It can be defined by two functions: the probability density function and the cumulative distribution function.
Probability distribution helps us in predicting the occurrence of an event easily. Let’s look at a few takeaways:
A probability distribution describes the predicted outcomes of various values for a specific data-generating process.
The factors that define a probability distribution are mean, standard deviation, skewness, and kurtosis.
Probability distributions come in various forms and have different properties.
To predict long-term returns on assets like stocks, probability distribution is used.
The essential properties of Probability Distribution are:
Every possible outcome has a probability of higher than or equal to zero.
The sum of all probabilities of all possible outcomes equals one.
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Discrete Probability Distribution vs. Continuous Probability Distribution
The two basic forms of probability distributions that describe distinct types of random variables are discrete and continuous probability distributions. It is important to understand their key differences, as this may help in data analysis.
Discrete Probability is utilized when a random variable can take specific, countable values. A discrete random variable may take a countable number of unique values, such as 0, 1, 2, 3...
Discrete probability distributions are used primarily in instances where only a countable number of outcomes can be listed.
For example: the number of customers visiting a company in an hour, or the probability of getting tails in five coin tosses.
On the other hand, continuous probability distributions are used to describe continuous random variables, where an infinite set of numbers within a certain range can be applied. These numbers are uncountable, as each interval contains endless possibilities. For example: The exact time required to accomplish a task.
How does Probability Distribution Work
Probability distribution represents how the probabilities of a random variable can be distributed, which helps in predicting the possible outcomes.
The normal distribution, also known as bell curve, is one of the most commonly used probability distributions.
The type of probability distribution a dataset belongs to is based on its data generation process, such as the probability density function (PDF) and the probability mass function (PMF).
What are the Types of Probability Distributions
Probability distributions are of two types. Let’s learn how they differ:
Discrete
A discrete probability distribution represents the probability distribution of a random variable that is finite.
- Binomial distribution
Used when an event occurs several times in a set number of trials.
Each trial has two possible results (such as present or absent, heads or tails).
Example: The likelihood of getting a head in 10 flips.
- Poisson Distribution
Models the probability of a certain number of events happening within a time or location, assuming the events happen independently at a steady average pace.
Examples include the number of customers that arrive at a bank each hour and the number of emails received per day.
Continuous
A continuous probability distribution displays the probability distribution of a random variable that is infinite.
- Normal distribution
A bell-shaped curve is described by its mean and standard deviation.
It is symmetrical, has no skew, and adheres to the 68-95-99.7% rule (data inside 1, 2, and 3 standard deviations). Example: Individual heights and stock market returns.
Real-life applications of Probability Distribution
We now understand the significance of probability distributions in mathematics. Let’s explore how they play a pivotal role in other fields beyond math:
- Normal distribution is quite often used to predict a student’s performance.
- Students can use the binomial distribution to predict their chances of getting correct answers on tests.
- Binomial distributions are widely used in medical fields to determine the effectiveness of medicines.
- Probability distributions are used in predicting customer behavior based on the trends they follow.
- Poisson distributions play an important role in cybersecurity by detecting fraudulent attempts.
Common Mistakes and How to Avoid Them in Probability Distribution
Students might make mistakes when solving probability distribution problems. These mistakes can be avoided with a proper understanding of the concepts and solutions. Let’s look at some common errors along with their solutions:
Mistake 1
Confuse Discrete with Continuous Distributions
Always ensure that PMF is used for discrete distributions such as binomials and PDF is primarily applied for continuous distributions such as normal and exponential distributions.
Mistake 2
Inconsistencies in Probability Table Construction
Keep in mind that the total sum of all probabilities in a discrete probability distribution will always be 1. Also, ensure that each probability value is appropriately allocated to its corresponding random variable.
Mistake 3
Confusing Expected Value and Variance
It is important to note that the expected value is a weighted average of all the possible outcomes, it is not necessarily the most likely one .
Mistake 4
Misunderstanding Conditional Probabilities in Distributions
Always apply the correct formula to obtain accurate results.
Correct formula: P(A|B) = P(A ∩ B)/ P(B).
Mistake 5
Calculation Errors
Do not round off values until the final step, and always check if the calculations are correct.
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Solved examples of Probability Distribution
Problem 1
A fair coin is tossed 6 times. What is the probability of getting exactly 5 heads?
The probability of obtaining 5 heads in 6 tosses is 0.093 which is equal to 9.375%.
Explanation
We have n= 6; probability of obtaining heads in just one toss: p = 0.5
Here, we apply the binominal probability formula:
P (X = x) =n! / x! (n - x) ! . (p)x. (q)(n-x)
Here, p (X = x) is the probability of obtaining 5 heads in 6 trials; x =number of trials in total and q = (1 – P) is the probability of getting tails.
Substituting the given values:
P (5) = 6! / 5! (6 - 5) ! = (0.5)5. (0.5)(6 – 5)
Now, we calculate the binomial coefficient:
6! / 5! (1) ! = 6! × 5! / 5! (1) ! = 6 / 1 = 6
Calculating the probability:
P (5) = 6 × (0.5)5 × (0.5)1
= 6 × 1/ 32 × 1 / 2
= 6 × 1 / 64 = 6 / 64 = 0.093
Problem 2
A student attempts 20 multiple-choice questions, each with five options. What is the probability of answering exactly 8 correctly if they guess randomly?
The likelihood of 8 right answers if they guess randomly is 0.00222 which is equal to 2.22%.
Explanation
Using the binomial probability formula:
P (X = x) = n! / x! (n - x) ! . (p)x. (q)(n-x)
Where,
n = 20
x = 8
p = 1/5
q = 1 – p = 4/5
Calculating the binomial coefficient:
20! / 8! (20 - 8) ! = 20!8! 12!
We now expand the first 12 terms of 20! To cancel with 12!:
(20 × 19 × 18 × 17 × 16 × 15× 14 × 13)8!
Calculating the probability:
(1/5)8 = 18/ 58 = 1/ 390625
= 0.00000256
(4/5)12 = 16777216/ 244140625 = 0.0687
Combining each part by multiplying them:
P(8) = 12,650 × 0.00000256 × 0.0687
= 12,650 × 0.000000176 = 0.032384
Now,
0.032384 × 0.0687 = 0.00222
Therefore, the probability is 0.00222.
Problem 3
A basketball player has a 40% chance of making a free throw. If they attempt 10 free throws, what is the probability that they make exactly 4?
The probability of making 4 free throws is 0.2508 which is equal to 25.08%.
Explanation
We use the formula for binomial probability:
P (X = x) =n!x! (n - x) ! . (p)x. (q)(n-x)
We have:
n = 10
x = 4
p = 0.4
q = 1 – p = 0.6
First, we compute the binomial coefficient:
n! / x! (n - x) ! = 10! / 4! (10 - 4) ! = 10! / 4! 6!
Here, we expand the factorials of 10:
= (10 × 9 × 8 × 7) / 8! = 5040/24 = 210
Now, compute the probability:
p4= (0.4)4 = 0.0256
q6 = (0.6)6 = 0.046656
Combining each part by multiplying them:
P(4) = 210 × 0.0256 × 0.046656
= 210 × 0.001194
= 0.2508
Problem 4
A factory produces 10 light bulbs, and each has a 3% chance of being defective. What is the probability that exactly 2 bulbs are defective?
The probability of 2 defective bulbs is 0.03195 or 3.195%.
Explanation
n = 10
x = 2
p = 0.03
q = 1 – 0.03 = 0.97
First, we compute the binomial coefficient:
n! / x! (n - x) ! = 10! / 2! (10 - 2) ! = 10! / 2! 8!
Expand the factorials:
= (10 × 9)2! = 902 = 45
Now, we compute the probability:
P2 = (0.03)3 = 0.0009
q8 = (0.97)8= 0.7890
Combining each part by multiplying them:
P(2) = 45 × 0.0009 × 0.7890
= 45 × 0.0007101
= 0.03195
Problem 5
In a class of 26 students, each has an 80% chance of passing a test. What is the probability that exactly 20 students pass?
The probability of 20 students passing the test is 0.169 which is equal to 16.9%.
Explanation
n = 26
x = 20
p = 0.80
q = 1 – 0.80 = 0.20
First, we compute the binomial coefficient:
n! / x! (n - x) ! = 26!20! (26 - 20) ! = 26!20! 6!
Expand the factorials:
= (26 × 25× 24 × 24 × 23× 22× 21 ) / 6!
= 165242320/ 720
= 229574
Computing the probability:
P20 = (0.80)20 = 0.0115
q6 = (0.20)6 = 0.000064
Combining each part by multiplying them:
P(20) = 229574 × 0.0115 × 0.000064
= 229574 × 0.000000736
= 0.1690
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FAQs on the Probability Distribution
1.What do you mean by a probability distribution?
2.What are the different kinds of Probability Distributions?
3.What is the main difference between Probability Mass Function (PMF) and Probability Density Function (PDF)?
4.Give the formula used for binomial probability distribution.
5.Can Probability Distribution be used in real life?
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Jaipreet Kour Wazir
About the Author
Jaipreet Kour Wazir is a data wizard with over 5 years of expertise in simplifying complex data concepts. From crunching numbers to crafting insightful visualizations, she turns raw data into compelling stories. Her journey from analytics to education ref
Fun Fact
: She compares datasets to puzzle games—the more you play with them, the clearer the picture becomes!
