Hey Developer’s, I’m back with a new topic which is Bayes Theorem in the series of statistics foundations.
Law of Total Probability:
If B1,B2,B3,⋯ is a partition of the sample space S, then for any event A we have,
So, let’s get started …
It is a way of finding a probability when we know certain other different probabilities. Basically calculates the probability of an event, based on the prior knowledge of conditions that might be related to the event.
where A and B are events and P(B) ≠ 0.
- P(A|B) is a conditional probability, the likelihood of an event A occurring given that B is true.
- P(B|A) is a conditional probability, the likelihood of an event B occurring given that A is true.
- P(A) and P(B) are the probabilities of A and B respectively.
Example Of Bayes Theorem
Example 1 : How search engines provides us the result ?
If we search for “Mobile Phones Under 10K”. Does the search engine know the mobiles under 10K ? No, but it knows from a lot of other searches what people are probably looking for and it calculates this on the basis of Bayes Theorem and shows us the relevant content.
Example 2 : Thanks to mathisfun for this example.
Let us say P(Fire) means how often there is fire, and P(smoke) means how often we see smoke, then:
P(Fire | Smoke) means how often there is fire when we can see smoke.
P(Smoke|Fire) means how often we can see smoke when there is fire.
So the formula kind of tells us “forwards” P(Fire|Smoke) when we know “backwards” P(Smoke|Fire).
- dangerous fires are rare (1%)
- but smoke is fairly common (10%) due to barbecues,
- and 90% of dangerous fires make smoke
We can then discover the probability of dangerous Fire when there is Smoke:
P(Fire|Smoke) = (P(Fire)P(Smoke|Fire))/P(Smoke) = (1% * 90%)/10% = 9%
So it is still worth checking out any smoke to be sure.
Example 3: Picnic Day
You are planning a picnic today, but the morning is cloudy
- Oh no! 50% of all rainy days start off cloudy!
- But cloudy mornings are common (about 40% of days start cloudy)
- And this is usually a dry month (only 3 of 30 days tend to be rainy, or 10%)
What is the chance of rain during the day?
We will use Rain to mean rain during the day, and Cloud to mean cloudy morning.
The chance of Rain given Cloud is written P(Rain|Cloud)
So let’s put that in the formula:
P(Rain|Cloud) =( P(Rain) P(Cloud|Rain)) / P(Cloud)
- P(Rain) is Probability of Rain = 10%
- P(Cloud|Rain) is Probability of Cloud, given that Rain happens = 50%
- P(Cloud) is Probability of Cloud = 40%
P(Rain|Cloud) =( 0.1 x 0.5)/ 0.4 = .125
Or a 12.5% chance of rain. Not too bad, let’s have a picnic!
Next Post will be on Random Variables and Discrete Random Variables
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