165 LearnersLast updated on September 6, 2025
Discrete Probability Distribution
Probability is the possibility that an event will occur. A discrete probability distribution describes the likelihood of outcomes for individually countable variables. In this topic, we will learn more about discrete probability distribution, its types, application, and many more.
What is a Discrete Probability Distribution?
The discrete probability distribution is a simple way to show the different chances of different outcomes when you can count them one by one. It gives the different values of a random variable along with its different probabilities. A discrete probability distribution contrasts with a continuous distribution, where outcomes can take any value along a continuum, as the outcome falls anywhere on a continuum. The types of discrete probability are Binomial, Poisson, and Bernoulli distribution. If a probability distribution is said to be a discrete probability distribution, it should follow these two conditions :
- 0 ≤ P(X = x) ≤ 1, which means the discrete random variable, X should be on the exact value, x, and it should be between 0 and 1
- ΣP (X = x) = 1, that is, the sum of all probabilities should be 1.
What are the Types of Discrete Probability Distributions
Now let’s discuss the types of discrete probability. The types are;
- Binomial
- Bernoulli
- Multinomial
- Poisson
- Binomial: Binomial probability is where there is a probability of getting two outcomes. Here after repetitive trails the data is collected in one of the two forms, into either success or failure. For instance, when flipping a coin, the probability of getting heads.
- Bernoulli: It is similar to binomial probability, as it also has two possible outcomes, with only one trial. Even here, the possible outcomes will be success or failure.
- Multinomial: In multinomial distribution, the probability of getting more than two outcomes with multiple counts. For example, a bowl filled with three different colored coins like red, blue, and green.
- Poisson: It is a type of probability that tells how a certain number of events happen in a fixed amount of time. For example, a Poisson distribution can be used for counting how many times a bus arrives at a stop in a fixed amount of time.
How to Calculate Discrete Probability Distribution
Here, we will discuss how to calculate a discrete probability distribution. To find a discrete probability distribution, the probability of mass function is required. Follow these steps to find the probability;
Step 1: Identify the sample, set of all possible outcomes of an experiment.
Step 2: Then you have to find the discrete random variable (x). It is a function assigning a numerical value to each outcome.
Step 3: Identifying the possible value of X.
Step 4: Finding the probability of each outcome by using the formula;
P (X = x) = Number of favorable outcomes/Total number of possible outcomes.
Step 5: Use a table to organize the results
Discrete Distribution vs. Continuous Distribution
Now let’s learn the difference between the discrete and continuous distribution.
|
Discrete Distribution |
Continuous Distribution |
|
The probability distribution for countable values |
The probability distribution for measurable values |
|
Here, the type of variable is discrete |
Here, the type of variable is continuous |
|
The graph of discrete distribution is a bar |
The graph of continuous distribution is a curve |
Real-life applications of Discrete Probability Distribution
In real-life discrete probability distribution are used to find the probability for countable values, now lets few real-world applications of it;
- For quality control in manufacturing, we use binomial distribution to find the number of defective products in a batch based on the random checks.
- When playing the games of chances, like for example cards, snake and ladder, etc. Probability helps you figure out how likely you are to win.
- The number of calls received in a call center is predicted using the Poisson distribution
- To identify whether an email is a spam or not using the Bernoulli distribution
Common mistakes and How to Avoid Them in Discrete Probability Distribution
Now let’s learn a few common mistakes and ways to avoid them to master discrete probability distribution.
Mistake 1
Not listing the sample space correctly
Not listing all possible outcomes leads to errors, so to avoid trying to use a list or a tree diagram to check whether the sample space is both correct and complete.
Mistake 2
Not checking whether the sum of the probabilities is 1
The sum of the probability for every valid probability is 1. To verify whether it is a correct answer or not, we need to add up and check whether the sum is 1 or not.
Mistake 3
Not considering the independence or dependence in events
Many formulae assume that events are independent and think that using them when events influence each other will yield errors. So analyze the events carefully and, if events are dependent, and choose the correct probability.
Mistake 4
Misidentifying the random variable X
If the value of X is not clear, it will lead to error. So clearly state find the value of X represent and list all the possible values to ensure it matches the experiment’s outcome
Mistake 5
Confusing Binomial and Bernoulli Distribution
As both the binomial and Bernoulli distribution the outcome are two, so students used to be confused with both. Students should understand where and when these are used.
Solved Examples of Discrete Probability Distribution
Problem 1
A factory produces light bulbs, and 5% of them are defective. If a random sample of 8 bulbs is taken, what is the probability that exactly 2 bulbs are defective?
The probability of getting 2 bulbs are 0.0515 or 5.15%
Explanation
Here, to find the probability we use the equation, P(X = k) = \(\binom{n}{k}\) pk(1 - p)n-k
According to the problem,
n = 8
k = 2
p = 0.05
Substituting the values,
P(X = 2) = \(\binom{8}{2}\) (0.05)2(1 - 0.05)8-2
P(X = 2) = \(\binom{8}{2}\) (0.05)2( 0.95)6
The value of C \(\binom{8}{2}\)= 28, (0.05)2 = 0.0025, and (0.95)6 = 0.7358
So, the value of P(X = 2) = 28 × 0.0025 × 0.7358 = 0.0515.
Problem 2
A basketball player has a 40% probability of making a free throw. What is the probability that their first successful shot happens on the third attempt?
The probability here is 0.144 or 14.4%
Explanation
Here the probability is of geometric distribution
So, P(X = k) = (1 - p)(k-1) × p.
Here, p = 0.4 and k = 3
Therefore, P(X = 3) = (0.6)2 × 0.4
= 0.36 × 0.4 = 0.144
Problem 3
A deck has 52 cards, including 4 Aces. If 5 cards are drawn randomly, what is the probability of getting exactly 2 Aces?
Probability ≈ 0.0399 or 3.99%.
Explanation
Here, the probability is calculated using \(P (X = k) = \dfrac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}}.\)
Here,
N = 52
K = 4
k = 2
n = 5
So P(X = 2) =\( \dfrac{\binom{4}{2} \binom{52 - 4}{5 - 2}}{\binom{52}{5}}.\)
P(X = 2) = 0.0399 or 3.99%.
Problem 4
A multiple-choice quiz has 10 questions, each with 4 answer choices. A student guesses randomly on each question. What is the probability of getting exactly 3 correct answers?
The probability is 0.250 or 25%
Explanation
Here, to find the probability we use the equation, P(X = k) = C \(\binom{n}{k}\) pk (1 - p)(n - k)
According to the problem,
n = 10
k = 3
p = ¼ = 0.25
Substituting the values,
P(X = 3) = \(\binom{10}{3}\)(0.25)2(1 - 0.25)10 - 3
P(X = 3) = \(\binom{10}{3}\) (0.25)2(0.75)7
Here, C\(\binom{10}{3}\) = 120
0.252 = 0.015625
0.757 = 0.1335
P(X = 3) ≈ 120 × 0.015625 × 0.1335 ≈ 0.250.
Problem 5
A factory produces screws with a 2% defect rate. What is the probability that the first defective screw appears on the 5th inspection?
The probability is 0.01845 or 1.845%
Explanation
P(X = 5) = (1 - p)(k-1) × p.
Here, p = 0.02
K = 5
P(X=5) = (1 − p)(k-1)p = (0.98)4 × 0.02
As 0.984 = 0.9224
P(X = 5) = 0.9224 × 0.02 = 0.01845
FAQs on Discrete Probability Distribution
1.What is a discrete probability distribution?
2.What is the difference between discrete probability distribution and continuous probability?
3.What are the real-life applications of discrete probability distribution?
4.What are the types of discrete probability distribution?
5.What are the conditions that a discrete probability distribution follows?
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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
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