Conditional likelihood is healthier than likelihood ; IF you’ve got the related data

Introduction

Anybody who has ever studied likelihood has heard of the long-established definition of likelihood of “Chance might be outlined because the variety of favorable outcomes divided by the entire quantity outcomes.” I can nonetheless hear my 4th grade instructor reiterating this!

Whereas this definition is appropriate, it typically makes me marvel, how correct is that this definition for the real-world? How correct is it when we now have some extra data concerning the favorable outcomes? To place it extra clearly, when we now have some extra “situations” to place to our favorable outcomes.

Placing “situations” is like slicing your unique pie of variety of favorable outcomes in a variety of methods, utilizing a number of situations, to provide the slice that represents the variety of favorable outcomes for you extra really. The picture beneath makes an attempt to depict this idea in a really transient method.

What higher approach than to symbolize what’s within the thoughts’s of lots of worldwide knowledge science college students learning within the US and searching for jobs! The unique variety of obtainable jobs is depicted on the acute left.

1st Situation : Introducing the 1st situation for “Work Expertise Length” refines the slice for the variety of obtainable jobs for a brand new joiner.

2nd Situation : Moreover, introducing the 2nd situation for “Nationality/Citizenship” refines the slice much more.

third Situation : The small darkish blue slice within the excessive proper chart represents essentially the most correct illustration for the variety of obtainable jobs (variety of favorable outcomes).

Earlier than continuing into why conditional likelihood could also be higher than likelihood, let’s do a fast recap of the definitions.

2. Definition of likelihood and conditional likelihood

Chance :

P(A) = Variety of favorable outcomes for A / Whole variety of outcomes

Conditional likelihood :

Now think about two occasions A and B. The muse of conditional likelihood is when there’s an “occasion given one other occasion”. On this case, when one says A given B, what meaning is the occasion A occurring given occasion B has occurred. So that’s attaching the “situation of B, to A”.

P(A|B) = P(A intersection B) / P(B) the place

P(A intersection B)* is given because the likelihood of each occasion A and occasion B occurring.

*given (A intersection B) and (A’ intersection B) are mutually unique. Therefore (A intersection B) union (A’ intersection B) = B.

After defining these barely complicated definitions, I’ll transfer to why I feel conditional likelihood is definitely higher.

3. Instance — Inspiration for this text

To begin off, the rationale I bought an thought for this text was once I was watching a Bollywood film the opposite day and when there was a scene about two previous pals discussing the likelihood of them bumping into one another!

Let me introduce some extra details about this scene:

• 1st pal : Police officer, initially from Mumbai metropolis, who was touring to Kalimpong; a small city, for a case.
• 2nd pal : Math professor, who was a resident of the city — Kalimpong.

These pals know one another since they each studied within the similar college.

• At the moment, the chums met one another at a restaurant the place the professor would go on a regular basis.

After introducing this data, let’s return to what was the likelihood of each of them bumping into one another in Kalimpong.

Police Officer : “Bro, what are the possibilities!”

Math Professor : “One out of 95675”

Police Officer : “Incorrect! You didn’t rely me”

Math Professor “I did. The present inhabitants is 95674”

Hmm… so let’s break this logic :

Preliminary Chance Calculation:

• The maths professor calculated the likelihood of assembly his pal, the police officer as 1/95,675.
• This assumes that every one the 95,674 residents of Kalimpong have the identical likelihood of assembly the professor because the police officer.

Why is that this calculation inaccurate:

• This calculation assumes that assembly the police officer is identical as assembly ANY OTHER resident of Kalimpong!

Introducing conditional likelihood:

Let’s think about some particular situations

I. Related Data:

• The police officer is a resident of Mumbai who traveled to Kalimpong.
• The maths professor goes to this cafe day by day.
• The police officer occurred to go to the identical cafe this one time.

II. Conditional Occasions:

Occasion A: The professor and police officer meet in Kalimpong.

Occasion B: The police officer travels from Mumbai to Kalimpong .

Chance of the 2 pals assembly :

1. The likelihood of police officer touring from Mumbai to Kalimpong, will depend on components resembling:

• How typically does he journey for work?
• How typically does he get assigned to circumstances from small cities?
• Let’s assume this likelihood is 0.1%.

2. The likelihood of the 2 pals assembly, will depend on components resembling:

• How typically do they each go to the cafe?
• How in style is the cafe?
• The professor frequently goes to the cafe.
• Let’s assume this likelihood is 1%.

Remaining calculations :

• The likelihood of the 2 pals assembly in Kalimpong, provided that the police officer is there, is 0.001%.
• It is a simplistic illustration of the idea, however what I’m attempting to say is to at all times search for extra related data to refine your likelihood.

Conclusion

Chance is easy and sophisticated on the similar time! Nonetheless there’s at all times refining we will do with any further data that we’re supplied. In actual world conditions, at all times attempt to search for how further data might help you add situations to make your chances extra correct.

Thanks for studying and I hope this text was helpful to you!