Q 3A: In Maharashtra, there are 20 Big companies in the IT sector. Out of these, 15 companies are in Mumbai. What is the probability that at least two companies are from Mumbai if a total of 3 companies are selected randomly to form a sample? (5 Marks) Note: Do not use any software for the calculation, you are advised to show all necessary steps of calculations.
In Maharashtra, there are 20 big corporations in the IT zone. Out of these, 15 companies are in Mumbai. What is an opportunity that at least two companies are from Mumbai if the three groups are selected randomly to shape a sample? (5 Marks)
Solution
To solve this problem the use of the binomial distribution, we can define the following parameters:
p = Probability of success (selecting a company from Mumbai) = 15/20 = 0.75
n = Number of trials (selecting 3 companies)
x = Number of successes (at least 2 companies from Mumbai)
q = Probability of failure (selecting a company not from Mumbai) = 1 – p = 1 – 0.75 = 0.25
We want to find the probability of getting at least 2 successes (x ≥ 2) out of 3 trials.
P(x ≥ 2) = P(x = 2) + P(x = 3)
Using the binomial probability formula:
P(x = k) = C (n, k) * p^k * q^ (n-k)
Where C(n, k) is the combination formula, given by:
C(n, k) = n! / (k! * (n-k)!)
Now let’s calculate the probabilities:
P(x = 2) = C (3, 2) * (0.75) ^2 * (0.25) ^ (3-2) = 3 * 0.5625 * 0.25 = 0.4219
P(x = 3) = C (3, 3) * (0.75) ^3 * (0.25) ^ (3-3) = 1 * 0.4219 * 1 = 0.4219
P(x ≥ 2) = P(x = 2) + P(x = 3) = 0.4219 + 0.4219 = 0.8438 or 84.38%
Interpretation: The probability that at least two companies in a randomly decided pattern of three businesses are from Mumbai is about 84.38% of the usage of the binomial distribution. This shows a high likelihood that after selecting three companies from the 20 big companies in Maharashtra, at least two could be from Mumbai.
This result means that Mumbai has a significant presence within the IT zone among the big businesses in Maharashtra. The possibility of selecting at least companies from Mumbai shows that the town is a central hub for IT companies, with a strong representation inside the sample. It demonstrates the concentration and prominence of IT companies in Mumbai compared to other areas in Maharashtra.
This information can be valuable for numerous functions, such as knowledge of the regional distribution of IT companies, planning business strategies, assessing competition, and identifying capacity possibilities inside the industry. Organizations interested in establishing a presence in Maharashtra’s IT sector may recollect Mumbai as a top area based on its higher representation within the sample. Moreover, policymakers and industry stakeholders can use this possibility to gauge the importance of Mumbai’s IT zone and its impact on the overall economy of Maharashtra.
But it’s important to note that the calculation assumes a random selection procedure based on the given opportunity values. The actual outcomes may range in practice, and other factors now not considered in this evaluation should affect the selection of companies. Therefore, while probability presents valuable insights, it should be considered alongside other relevant factors and facts for making informed decisions.
Note: The above solution is only for reference purposes, Do not copy any answers or it will lead to the rejection of the paper.
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