Probability Distribution Definition Formula With Examples

What is the Probability Distribution?

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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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 example, possible outcomes could be (H, H), (H, T), (T, H), and(T, T).

Then possible no. of heads selected will be – 0, or one could calculate 1 or 2, and the probability of such an event 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: The event was ‘No. of heads’ in the given example. 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 in the hall. So, what will be the probability distribution of ‘selecting at least one woman.’

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
=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
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. They decide to select randomly for the final selection, and the number of women selected could be either 0 or 1, or 2. The possibility of an event where no women would be selected is, and the possibility of an event where it will select only 1 woman amounted to. In contrast, the possibility of selection of both women is.

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

Example #3

In a similar type of situation, let’s assume a situation where a manufacturing company named ABC Inc. was engaged in manufacturing tubelights. 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, they produced 10 tube lights, out of which 2 were damaged. The Manager decided to pick 3 of the tubelights randomly. Prepare the probability distribution of selecting damaged goods.

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

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

So, one could calculate the probability 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 1st round X probability of selecting good light in 2nd round X probability of selecting good light in 3rd round.

P (0) = P(G) X P(G) X P(G)= 8/107/96/8 = 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 1st round or 2nd or 3rd round)

So,

P (1) = (8/107/92/8)*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 1st round or 2nd or 3rd round)

P (2) = (8/102/91/8)*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 by randomly selecting goods and evaluating the chances of producing damaged goods.

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

Relevance and Uses

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

What is the Probability Distribution?#

Probability Distribution Formula#

Examples of Probability Distribution Formula (with Excel Template)#

Example #1#

Example #2#

Example #3#

Relevance and Uses#

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