Probability Distribution

What is the Probability Distribution?

Probability distribution could be defined as the table or equations showing respective probabilities of different possible outcomes of a defined event or scenario. In simple words, its calculation shows the possible outcome of an event with the relative possibility of occurrence or non-occurrence as required.

Probability Distribution Formula

The probability of occurring event can be calculated by using the below formula;

Probability of Event = No of Possibility of Event / No of Total Possibility

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Source: Probability Distribution (wallstreetmojo.com)

Examples of Probability Distribution Formula (with Excel Template)

Given below are the examples of the probability distribution equation to understand it better.

Example #1

Let’s suppose a coin was tossed twice, and we have to show the probability distribution of showing heads.

Solution

In the given an example, possible outcomes could be (H, H), (H, T), (T, H), (T, T)

Then possible no. of heads selected will be – 0 or 1 or 2, and the probability of such event could be calculated by using the following formula:

Calculation of probability of an event can be done as follows,

Using the Formula,

Probability of selecting 0 Head   = No of Possibility of Event / No of Total Possibility

• = 1/4

Probability of an event will be –

• =1/4

Probability of selecting 1 Head   = No of Possibility of Event / No of Total Possibility

= 2/4

= 1/2

Probability of selecting 2 heads  =No of Possibility of Event / No of Total Possibility

= 1/4

So, the probability distribution for selecting heads could be shown as;

Explanation: In the given an example, the event was ‘No. of heads’. And the number of heads that can occur is either 0 or 1 or 2, which would be termed as possible outcomes, and the respective possibility could be 0.25, 0.5, 0.25 of the possible outcomes.

Example #2

In an interview hall, there were 4 people present consisting of 2 men and 2 women after being tested by the interviewers. But the concerned company had only 2 vacancies to fill. So the interviewer decided to select 2 candidates from the people present in the hall. What will be the probability distribution of ‘selecting at least one woman.’

Solution

In the given case, the number of possibilities of selecting candidate could be,

(W1, W2), (W1, M1), (W1, M2), (W2, M1), (W2, M2), (M1, M2)

As per the requirement, let’s denote the event ‘number of women’ as X, then the possible values of X could be;

X = 1 or 2

Calculation of Probability of an event

• So, the probability of selecting 0 women = no of the possibility of selecting 1 women / total possibilities

Probability of an event will be –

• =1/6

Similarly,

Probability of selecting X women = no of the possibility of selecting X women / total possibilities

• So, the probability of selecting 1 woman = no of the possibility of selecting 1 women / total possibilities
• =4/6
• =2/3

Similarly,

• Probability of selecting 2 women =no of the possibility of selecting 2 women / total possibilities
• =1/6

Now, as per the question, the probability of selecting at least 1 woman will be

• = Probability of selecting 1 woman + Probability of selecting 2 women
• = 2/3+1/6

• =5/6

So, the probability distribution for selecting women will be shown as;

Explanation: In this scenario, the management decided to fill up the 2 vacancies through interviews, and during the interview, they chose 4 people. For the final selection, they decide to select randomly, and the number of women selected could be either 0 or 1 or 2. Possibility of an event where no women would be selected is  & the possibility of an event where only 1 woman will be selected amounted to, whereas the possibility of selection of both women is.

So, through the use of Probability distribution, the trend of employment, trend of hiring, selection of candidates, and other nature could be summarised and studied upon.

Example #3

In a similar type of situation, let’s assume a situation where a manufacturing company named ABC Inc. was engaged in the manufacturing of tube lights. One day the Operation Manager decided to randomly evaluate the effectiveness of production by evaluating the percentage of Damaged stocks produced within 1 hour. Let’s say, within 1 hour, 10 tube lights were produced, out of which 2 were damaged. The manager decided to pick 3 of the tube lights randomly. Prepare the probability distribution of selecting damaged goods.

Solution

In the given example, the random variable is the ‘number of damaged tube lights selected.’ Let’s denote the event as ‘X.’

Then, the possible values of X are (0,1,2)

So, the probability could be calculated by using the formula;

Probability of selecting X  = no of possibilities of selecting  X / total possibilities

Then,

Probability of selecting 0 damaged lights = probability of selecting good light in 1^st round X probability of selecting good light in 2^nd round X probability of selecting good light in 3^rd round

• P (0) = P(G) X P(G) X P(G)
• = 8/10*7/9*6/8
•  = 7/15

Similarly, Probability of selecting only 1 damage light = [P(G) X P(G) X P(D)] X 3

(multiplied by 3 because the damaged light can be selected in 3 ways, i.e., either in 1^st round or 2^nd or 3^rd round)

So,

• P (1) = (8/10*7/9*2/8)*3
•  =7/15

Similarly, Probability of selecting 2 damage lights = [P(G) X P(D) X P(D)] X 3

(multiplied by 3 because the good light can be selected in 3 ways, i.e., either in 1^st round or 2^nd or 3^rd round)

So,

• P (2) = (8/10*2/9*1/8)*3
• = 1/15

So the probability of selecting at least 1 Damaged lights = Probability of selecting 1 Damage + Probability of selecting 2 Damage

• = P (1) + P (2)
• =7/15+1/15
• =8/15

So, the probability distribution for selecting damage lights could be shown as;

Explanation: The operation Manager of the Business organization wanted to evaluate the effectiveness of the process through the random selection of goods and evaluating the chances of production of damaged goods.

Through this example, we can see that the industry can also use Probability distribution for evaluating the effectiveness of its processes and the ongoing trends.

Relevance and Uses

A probability distribution is basically used for recording the possibility of occurrence or non-occurrence of a certain event. From a business point of view, it can also be used for predicting or estimating the possible future returns or profitability of the business. In modern-day business, the probability distribution calculation is used for sales forecasting, risk evaluation, finding and evaluating the obsolete part of any business or process, etc.

Recommended Articles

This has been a guide to Probability Distribution and its meaning. Here we discuss the formula to calculate probability distribution along with practical examples and a downloadable excel template. You can learn more from the following articles –

• Normal Distribution
• Exponential Distribution
• Poisson Distribution Meaning
• Lognormal Distribution Excel