Conditional Probability in statisics

Conditional Probability

A Conditional probability is a probability whose sample space has been limited to only those outcomes that fulfill a certain condition.

A rule that can be used to determine a conditional probability from unconditional probabilities is:

  • P(A | B) = Conditional probability that event A will occur given that event B has occurred already.
  • P(A ∩ B) = Unconditional probability that event A and event B both occur.
  • P(B) = Probability that event B occurs.
  • The usual notation for "event A occurs given that event B has occurred" is "A | B" (A given B).
  • The symbol | is a vertical line and does not imply division. 
For example, suppose you go out for lunch at the same place and time every Friday and you are served lunch within 15 minutes with probability 0.9.
However, given that you notice that the restaurant is exceptionally busy, the probability of being served lunch within 15 minutes may reduce to 0.7.
This is the conditional probability of being served lunch within 15 minutes given that the restaurant is exceptionally busy.

Lets Solve one situation by using Conditional Probability

Consider the college applicant who has determined that he has 0.80 probability of acceptance and that only 60% of the accepted students will receive dormitory housing. Of the accepted students who receive dormitory housing, 80% will have at least one roommate. 

The probability of being accepted and receiving dormitory housing and having no roommates is calculated by:
P(Accepted and Dormitory Housing and No Roommates) 
= P(Accepted) P (Dormitory Housing | Accepted) P (No Roomates|Dormitory Housing and Accepted) 
= (0.80)*(0.60)*(0.20) = 0.096. 
The student has about a 10% chance of receiving a single room at the college.

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