5.X.3. Derivation of Bayes’ Theorem

Try these first

1. 1.

   Define sample space.

   Hint

   Think of what you would observe.

2. 2.

   What does conditional probability measure?

   Hint

   It involves two events.

3. 3.

   What is Bayes' Theorem used for?

   • To derive frequency distributions
   • To update probabilities based on new data
   • To calculate means and variances

   Hint

   Think about updating beliefs.

4. 4.

   True or False: The prior probability changes when new evidence is introduced.

   • True
   • False

   Hint

   What does 'prior' mean?

5. 5.

   A factory produces 5% defective items. An inspection process has an 80% success rate in detecting defects. If an item is inspected and found defective, what is the probability that it is actually defective?

   Hint

   Work through joint probabilities and apply Bayes' Theorem carefully.

6. 6.

   In clinical trials, a new drug shows efficacy in 70% of patients. There is a 10% chance of side effects. If side effects occur, what's the updated probability that the patient benefited from the drug?

   Hint

   Set up your known probabilities and use Bayes' formula.