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The Complete Beginner's Guide to Subsets and Probability

Summary

We try to go over the below cases to learn and develop our thinking around the subsets in probability. Below excerpts are taken from the news of the day.

Problem 1

Let's consider the latest movie release 'Sanju' starring Sanjay Dutt in Bollywood, scored a century and surpassing the highest-single day record.
Which odds have higher probability?
A. Sanju is a hit film.
B. Sanju is a hit film and will help Sanjay Dutt in his image transformation.

Problem 2

Let's try the next challenge, with China targetting to 5G, Indian government modified the tender norms mandating the use of 2G technology for government-funded mobile service projects.
Which odds have higher probability?
A. Percentage Regions with 4G Technology
B. Percentage Regions with 4G Technology and 2G Technology

Problem 3

Let's consider this, Indian I-T dept launches 'instant' Aadhaar-based online PAN allotment system because of an increasing number of people applying to obtain the Permanent Account Number (PAN) for their financial and tax matters.
Which odds have higher probability?
A. Receiving an ePAN
B. Receiving an ePAN when you have existing physical PAN

Explanation

All the problems above expands the concept of subsets in probability.

Let's consider having four offices

officename = {
"Horizontal Wing A", 
"Horizontal Wing B", 
"Techno Sector-1", 
"Techno Sector-2" 
}

Which odds have higher probability?
A. Office name contains Alphabets

officename = {
"Horizontal Wing A", 
"Horizontal Wing B", 
"Techno Sector-1", 
"Techno Sector-2" 
}

B. Office name includes Alphabets and number

officename = {
"Techno Sector-1", 
"Techno Sector-2" 
}

We can see in the above example that Event A includes all of the B's cases. Here, we say event B is the subset of event A and probability or likelihood of event B should be less than or equal to A.

Solution to Problem 1

The explanation for Problem 1, the Event B that states both Sanju is a hit film and will help Sanjay Dutt in his image transformation is a subset of the event that Sanju is a hit film. Thus it cannot be more likely than the event A that Sanju is a hit film.
The same solution applies to the other problems vis. Problem 2 and Problem 3

Mathemetical Formula

Regarding probability, if event X is a subset of event Y, then
P(X) <= P(Y)
The Complete Beginner's Guide to Subsets and Probability
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