Cub11k's BIU Notes

Probability 04

8 min read

Bayes formula and law of total probability

”Short” law of total probability lemma

”Short” Bayes formula definition

Example

Let there be some insurance company. The insurance company believes that all people are divided into two categories - accident prone ones and not accident prone ones. Accident prone ones are considered to be of population. Accident prone ones have probability of to have at least one accident in the next year. Not accident prone ones have probability of to have an accident in the next year. What is the probability of some new client having an accident in the next year?

Let us now assume that the new client had an accident last year. What is the probability of him being accident prone?

Example

Let there be a deck of n distinct cards. One player is opening cards one by one. Second player has an opportunity to call one card (the next one) to be a card . Is there a specific moment in which the second player should call the next card?

Example

Let there be a multiple choice question with possible answers Student knows the correct answer with probability In case the student doesn’t know the answer, he takes a guess, with probability he gives a correct answer. What is the probability of student actually knowing the correct answer if he answered correctly?

Example

Let there be some blood test for disease With probability it detects a disease if it is present (true positive) With probability it detects a disease if it is not present (false positive) of population have the disease What is the probability of a person having the disease if the test is positive?

Example

Professor studies disease Professor has a dilemma:

  • If he is at least certain that his patient has disease - he immediately performs the surgery
  • If he is less than certain - he asks to do some additional expensive testing Let there be a new diabetic patient, professor was certain that he has disease Patient took the test and got the positive result The test always gives true positives, and gives false positives Should the professor perform surgery or perform more tests?

Law of total probability theorem

Example

Let’s take a previous example of cards game There are cards, excluding card There are different orders of these cards

Bayes formula definition

Example

Let there be a plane is missing. There are three different disjoint areas - If the plane is in region , we can locate it with probability We tried to find it in area and we didn’t succeed. What is the probability of the plane being in area

Example

Let there be three cards, one with both sides being red, one with sides being black and one with one red and one black side. One card is randomly selected, the side we see is red What is the probability of the other side to be black?

Example

Let there be a small town A new family is moving into a house in this town Family is father, mother and two kids We can see a mother walking with her daughter What is the probability that both kids are girls?

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