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Conditional Probability

Conditional Probability. The idea. In general, the probability of an event is affected to some degree by the occurrence of other events. Example #1. A family with 2 children has moved onto your street. Let p(E) be the probability that the family has 2 boys. Find p(E). Example #2.

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Conditional Probability

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  1. Conditional Probability

  2. The idea • In general, the probability of an event is affected to some degree by the occurrence of other events.

  3. Example #1 • A family with 2 children has moved onto your street. • Let p(E) be the probability that the family has 2 boys. Find p(E).

  4. Example #2 • A family with 2 children has moved onto your street. A neighbor told you that he has met the couple whose youngest child is a boy. • Letp(E/F) be the probability that the family has 2 boys given that it is known that the couple’s youngest child is a boy. Find p(E/F).

  5. Remarks • p(E), found in Example #1, is the type of probability that we have been computing but p(E/F), found in Example #2, is what is called a Conditional Probability. • We will consider a third example so that we can formally define Conditional Probability.

  6. Example #3 • A group of 1000 people includes 100 employees of Montgomery College and 350 females. There are 50 female MC employees in the group. • Suppose that a person is selected at random from that group. • Let p(E) be the probability that the person selected is an employee of MC and • p(E/F)the probability that the person selected is an employee of MC given the person is female. • Compute p(E) and compute p(E/F).

  7. Definition • TheConditional Probability of E given Fis the likelihood of the occurrence of the event E if it is assumed or known that the event F had taken place. • As suggested by the remarks on the last example, it is defined by • Note: We divide by the probability of the event that is assumed to have happened.

  8. Example #4 • Use the given Venn diagram to find: • p(E/F) • p(F/E) • p(E’/F) • p(E’/F’)

  9. We now consider the use of the conditional probability formula in a real-life situation.

  10. Example #5 • A study of 200 samples of ground meats revealed that the probability that a sample was contaminated by salmonella was 0.20. • The probability that a salmonella-contaminated sample was contaminated by a strain resistant to at least three antibiotics was 0.53. • What was the probability that a ground meat sample was contaminated by a strain of salmonella resistant to at least three antibiotics?

  11. To grasp what we are given and what we are supposed to find, we first rephrase the problem: • The probability that a sample was contaminated by salmonella was 0.20. • The probability that a sample was contaminated by a strain resistant to at least 3 antibiotics given that it was contaminated by salmonella was 0.53. • We need to determine the probability that a sample was contaminated by salmonella and by a strain resistant to at least 3 antibiotics.

  12. Product Rule • More generally, if we multiply by p(F), the Conditional Probability Formula yields the Product Rule:

  13. Tree Diagrams • The method consisting of using a tree diagram is one with far-reaching effects. • We can show that to find the probability of an event using a tree diagram: • We simply need to multiply along all the branches leading to that event and add the results.

  14. Example #6 • There are 300 employees at your place of employment, of which 140 are males. • It is known that 80% of the males and 60% of the females have internet service in their homes. • If an employee is selected at random from your workplace, what is the probability that the employee is: • A female and has internet service at home? • A male who does not have internet service at home? • Has internet service at home?

  15. Remarks • The method consisting of using a probability tree diagram will become a powerful tool in our hands as we pursue the study of probability and its applications. • We will not need to remember a number of formulas. • Rather we will strengthen our skills of drawing and completing probability tree diagrams. • We will only need to commit to memory:

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