5.0. Bayes’ Theorem

Try these first

1. 1.

   What is the sample space?

   Hint

   Think about all the outcomes of a dice roll.

2. 2.

   What does P(A | B) represent?

   Hint

   It's the probability of an event under certain conditions.

3. 3.

   What does the prior probability represent in Bayes' Theorem?

   • Updated belief
   • Initial belief
   • Evidence

   Hint

   Consider the beliefs you have before receiving new information.

4. 4.

   True or False: In Bayes' Theorem, the posterior probability is always greater than the prior probability.

   • True
   • False

   Hint

   Reflect on the effect of evidence on beliefs.

5. 5.

   A rare condition affects 0.5% of the population. A screening test identifies the condition with 98% sensitivity and 90% specificity. Calculate the probability of having the condition given that a person tests positive.

   Hint

   Remember to first derive P(T) using total probability.

6. 6.

   A bag contains 10 blue and 20 red balls. You draw a ball at random and it’s red. What is the probability that the next drawn ball is red if you replace it?

   Hint

   Think about the events being independent due to replacement.