How to Use Probability Distribution Formula & Table in Exam Questions
The concept of probability distribution plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. It describes how likely different outcomes are during a random experiment, like tossing coins, rolling dice, or measuring heights. Understanding probability distributions helps students solve problems confidently in Class 12 Maths, competitive exams, and many logical reasoning situations.
What Is Probability Distribution?
A probability distribution is a mathematical function or table that assigns the probability to every possible outcome of a random experiment. In simple words, it shows how likely it is for each result to happen. You’ll find this concept applied in statistics, data analysis, weather prediction, and even in daily games of chance. Common uses include finding the probability in board exams, conducting surveys, or doing research.
Types of Probability Distributions
Probability distributions come in two main types – discrete and continuous:
| Type | What It Means | Example |
|---|---|---|
| Discrete Probability Distribution | Probabilities are assigned to specific, countable outcomes (whole numbers). | Number of heads in 5 coin tosses |
| Continuous Probability Distribution | Probabilities are spread over a range of possible values (can include fractions or decimals). | Heights of students in a class |
Key Formula for Probability Distribution
Here are the important formulas you should know:
| Distribution Type | Formula | Meaning |
|---|---|---|
| Discrete (Binomial) | \( P(X = r) = nCr \cdot p^{r} \cdot (1-p)^{n-r} \) | Chance of exactly r successes in n trials |
| Continuous (Normal) | \( P(x) = \frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \) | Probability curve for a variable, like height |
Where \( n \) = total trials, \( r \) = number of successes, \( p \) = probability of single success, \( \mu \) = mean, \( \sigma \) = standard deviation.
How to Make a Probability Distribution Table
You can use a table to organize all possible values a variable can take and their probabilities:
| Outcome (X) | Probability P(X) |
|---|---|
| 0 | 0.1 |
| 1 | 0.4 |
| 2 | 0.3 |
| 3 | 0.2 |
The total of all probabilities must always add up to 1.
Step-by-Step Illustration
Example: What is the probability of getting 2 heads if a coin is tossed 3 times? (Use binomial distribution)
1. Number of trials: n = 32. Number of successes (heads): r = 2
3. Probability of getting head in one toss: p = 0.5
4. Use formula: \( P(X=2) = 3C2 \cdot (0.5)^2 \cdot (1-0.5)^{3-2} \)
5. Calculate: \( 3C2 = 3 \), \( (0.5)^2 = 0.25 \), \( (0.5)^1 = 0.5 \)
6. Multiply: \( 3 \times 0.25 \times 0.5 = 0.375 \)
7. So, probability = 0.375
Speed Trick or Vedic Shortcut
When working with a probability distribution, check if probabilities in your table add up to 1. If not, there’s a mistake! For binomial, if n is small, use Pascal’s Triangle to find coefficients quickly instead of calculating nCr every time.
Exam Tip: For Normal Distribution, 68% of data falls within 1 standard deviation (σ) of the mean (μ); this can help with MCQ elimination.
Frequent Errors and Misunderstandings
- Forgetting that the sum of all probabilities in a distribution must be exactly 1.
- Confusing discrete (countable) and continuous (any value in an interval).
- Using wrong formula or incorrect values for n, r, or p in binomial problems.
Relation to Other Concepts
Knowing probability distribution is essential for mastering probability basics, Probability Mass Function (PMF), random variables, and Normal distribution. Mastery helps in tackling advanced questions in data handling, statistics, and science experiments.
Try These Yourself
- Create a probability distribution table for rolling a dice and getting 1, 2, 3, or 4.
- Identify whether ‘number of rainy days in a week’ is discrete or continuous.
- Solve: Find the probability of getting exactly 5 heads in 10 coin tosses.
- Explain why the probability of any event can never be negative.
Classroom Tip
To quickly spot the type, ask: “Can I list all outcomes (discrete), or could the answer be a decimal (continuous)?” Save examples in your phone as tables or diagrams for a quick pre-test revision. Vedantu’s teachers often build such visual study notes in live sessions.
We explored probability distribution—from definition, formulas, types, and hands-on examples, to common mistakes and connections to other maths topics. Continue practicing with Vedantu to boost your score in probability questions and become confident in using this concept for Class 12, JEE, and practical life!
Explore further: Probability | Probability Density Function | Probability Questions
FAQs on Probability Distribution Explained: Formula, Types & Solved Examples
1. What is a probability distribution in simple terms for a Class 12 student?
A probability distribution is a mathematical function or table that describes the likelihood of all possible outcomes for a random variable in an experiment. In simpler terms, it provides a complete picture of the probabilities associated with every possible result, such as the chances of getting 0, 1, 2, or 3 heads when a coin is tossed three times.
2. What is a random variable and how is it connected to a probability distribution?
A random variable, usually denoted by X, is a variable whose value is a numerical outcome of a random phenomenon. For example, if you roll a die, the random variable X could be the number that appears on top. The probability distribution is what gives this random variable meaning by assigning a specific probability to each of its possible numerical values (e.g., P(X=1) = 1/6, P(X=2) = 1/6, and so on).
3. What is the main difference between a discrete and a continuous probability distribution?
The main difference lies in the type of outcomes the random variable can take:
- A discrete probability distribution deals with outcomes that are countable and have distinct values. For example, the number of heads in four coin tosses (can be 0, 1, 2, 3, or 4).
- A continuous probability distribution deals with outcomes that can take any value within a given range. These are typically measurements. For example, the height of students in a class, which could be 165.1 cm, 170.25 cm, etc.
4. What is a binomial distribution and when should it be used in solving problems?
A binomial distribution is a specific type of discrete probability distribution used when an experiment, known as a Bernoulli trial, is repeated a fixed number of times. You should use it only when these four conditions are met:
- There is a fixed number of trials (n).
- Each trial is independent of the others.
- Each trial has only two possible outcomes: 'success' or 'failure'.
- The probability of success (p) remains constant for each trial.
5. What is the formula for the binomial probability distribution as per the CBSE syllabus?
The formula to calculate the probability of getting exactly 'r' successes in 'n' trials is: P(X = r) = nCr × pʳ × qⁿ⁻ʳ. Here is what each variable represents:
- n is the total number of trials.
- r is the specific number of successful outcomes.
- p is the probability of success in a single trial.
- q is the probability of failure in a single trial, which is always calculated as (1 - p).
- nCr is the combination formula for calculating the number of ways to choose 'r' successes from 'n' trials.
6. Why are the mean and variance important for understanding a probability distribution?
The mean and variance are crucial because they summarise the key characteristics of a distribution in just two numbers. The mean (or expected value) tells you the long-term average outcome you can expect from the random experiment. The variance measures the spread or dispersion of the outcomes, indicating how much they typically deviate from the mean. A low variance means outcomes are clustered close to the average, while a high variance suggests they are more spread out.
7. What are the essential steps to follow when solving a problem involving a probability distribution?
To solve a probability distribution problem systematically, you should follow these steps:
- Step 1: Identify the random variable (X) and determine if it is discrete or continuous.
- Step 2: List all possible numerical values that the random variable can take.
- Step 3: Calculate the probability for each possible value of the random variable using the appropriate formula or logic.
- Step 4: Present the results in a probability distribution table.
- Step 5: Verify your calculations by checking that the sum of all probabilities equals exactly 1.
8. How can probability distributions be used to model real-world situations?
Probability distributions are essential for modeling uncertainty in many real-world fields. For example:
- In quality control, a binomial distribution can model the number of defective items in a batch produced by a factory.
- In insurance, a Poisson distribution can predict the number of claims an insurance company will receive in a month.
- In biology, a normal distribution can represent the distribution of physical traits like height or blood pressure in a population.
9. What is a common mistake students make when creating a probability distribution table?
A very common mistake is failing to verify that the sum of all probabilities equals 1. After calculating the probability for every possible outcome of the random variable, you must add them all up. If the total is not exactly 1, there is an error in your calculations, either in one of the probability values or because you missed a possible outcome for the random variable.
10. Why must the sum of all probabilities in any discrete probability distribution always equal 1?
The sum of all probabilities in a distribution must equal 1 because the set of all possible outcomes represents the entire sample space of the experiment. This means that when the experiment is performed, one of the outcomes in the distribution is absolutely certain to occur. In probability theory, an event that is certain to happen always has a probability of 1.
