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Theorem And Conditional Probability With Example

Referencing Styles : Harvard | Pages : 1

In the field of probability and statistics, Bayes’ theorem gives a brief description about the probability of a particular event related to prior knowledge of the probable conditions that might have a certain relevance to the event. For example, if a person is affected by cancer which is related to age, using Bayes’ theorem a person’s age can be utilized to estimate the probability of suffering through cancer being compared to the assessment of the probable chances of cancer without having the knowledge of the person’s age. On the other hand, Conditional probability is the measure of probability of an event which has already occurred in the past. The concept of probability is one of the most fundamental part having an existence within the probability theory. Conditional probabilities though can be quite slippery and shall require minute interpretation. For example, there necessarily needs to be a relationship between A and B and there is no need for their simultaneous occurrence.

For example, suppose that there is a test for using drug is 99% sensitive and 99% specific. That is, the test will have a production of 99% true positive results of the drug users and 99% true negative results for the non-drug users. Suppose that 0.5% of people are users of drugs. What shall be the probability that a randomly selected person with a positive test is a drug user?

 

Bayes’ theorem is defined mathematically with the following equation,

P (A|B) = P (B|A).P (A) / P (B)

Where, A and B are the events specified and P (B)! = 0.

  • P (A|B) is a conditional related to probability. The similar event to that of A given a condition that B is true.
  • P (B|A) is also an instance of conditional probability, similar to the one of event B occurring only if event A is true.
  • P (A) and P (B) are the particular probabilities of observing A and B independent to each other which is also referred to as marginal probability.

In the field of probability theory and its applications, Bayes’ theorem showcases the relation between the aspect of conditional probability and its reverse form. For example, the probability related to any hypothesis given as some pieces of evidences under observation and the probability of that evidence given under the light of that specific hypothesis.  

There is a cancer test including the implementation of the Bayes’ theorem, very much separate from the event of actually having a cancer. It covers the tests which actually detect things which do not exists and might miss out on things which actually do exist. Often, test results without the adjusting for test errors is placed for usage. Bayes’ theorem takes the responsibility to convert the results from the test into the real probability of the event. Bayes’ theorem also states that, if the real probabilities for the chance to have turned out as positive or negative then the measurement errors can be corrected, given the actual facts and figures are approximately correct and up to date.

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