Bayes' Theorem
Chapter Name : Probability |
Sub Topic Code : 104_12_13_05_02 |
Topic Name : Bayes' Theorem |
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Sub Topic Name : Theorem Of Total Probability |
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Introduction
• For computing probabilities of various events we use a divide and conquer approach called Total Probability Theorem. • Total probability theorem is used in conjunction with Bayes theorem. • Bayes theorem is a simple mathematical formula used to solve conditional probabilities.
Pre-Requisites:
• Knowledge of events in probability. • Knowledge of set theory.
Activity:
You will observe that by applying Total probability theorem we can calculate the percentage of bulb to work for more than 500 hours.
Real Life Question:
An insurance company rents 30% of cars for its customers from agency I and 60% of cars from agency II. If 7% cars of agency I and 4% cars of agency II breakdown during rental period, what is the probability that a rented car by this insurance company breaks down?
Key Words / FlashCards
| Key Words | Definitions (pref. in our own words) |
|---|---|
| Probability | Probability is the chance that something will happen. |
| Event | An event is a set of outcomes to which a probability is assigned. An event is subset of sample space. |
| Total probability Theorem | The law of total probability is a rule relating marginal probabilities to conditional probabilities. |
| Bayes Theorem | Bayes theorem shows relation between conditional probabilities and its reverse form. |
Learning aids / Gadgets
| Gadgets | How it can be used |
|---|---|
| Electric bulbs | Suppose that you bought two bulbs X and Y from market of two different companies. Factory X’s bulbs work for over 500 hours in 99% cases, whereas factory Y’s bulb work for over 500 hours in 95% cases. It is known from market that only 60% stock is available for factory X’s bulb. What is the chance that purchased bulb will work longer than 500 hours? |
Real life uses :
Bayes theorem is used in engineering, medicine, law.
Places to visit :
Suppose you have internet connectivity problem that’s been going on for days and you are not sure what’s causing it. You believe it could be caused by router (hypothesis A) or electricity problems (hypothesis B). You can apply Bayes rule here. What makes the rule so useful is that it tells you what questions you need to ask to evaluate how strong a piece of evidence is.
Practical examples around us
| Examples | Explainations |
|---|---|
| Bayes rule in medical sciences. | Doctor performs the test with 99% reliability that is 99% people who are sick test positive and 99% people who are healthy test negative. The doctor knows only 1% people in locality are sick. The question is if the patient tests positive what are the chances that the patient is sick? |
What you learn in Theory:
Bayes theorem shows relation between conditional probabilities and its reverse form.
What you learn in Practice:
Bayes rule implications make you better at figuring out the truth in many real life situations.