Text exercises For each exercise:

(a) identify the alternatives A_k;

(b) identify the pivotal event B;

(c) calculate the prior probabilities P(A_k) and the likelihoods P(B|A_k);

(d) calculate the posterior probabilities P(A_k|B).
i. You have 6 dice, D₁, D₂, D₃, D₄, D₅, and D₆. A friend of yours picks one of them at random. He tosses it, and gets 3. For each k, find the probability that the die he picked was the D_k.
ii. Some students have a biased D20. For this die:

• the probability of an even number is 9/15 ;
  the probability that the outcome is in the top six is 4/5;
• the conditional probability of an even number, given that the outcome is in the top six, is 2/3.
  If you know the die has landed on an even number, what is the probability that it is in the top six?
• iii. The proud nation of Molvania has 2 weightlifting teams. On one team (“the steroid team”), 70% use steroids, whereas on the other team (“the clean team”), 10% use steroids.
   What is the probability that a randomly picked lifter uses steroids? (The teams are equally large, so each lifter has the same chance of being picked.)
   To decide which team to send to the Olympics, Molvania’s minister of sports tosses a die, and the probability that he sends the steroid team is 2/3 . Now, we read in the newspaper that a Molvanian lifter was randomly picked for a steroid test during the Olympics, and his test came back negative. Use this information to update the probability that the steroid team was sent to the Olympics.
• iv. Genius Consulting are testing out 3 different statistics software bundles:
  Freequen, Gnormal and Halfling.When they test the software with a problem with 100 variables, Freequen crashes 10% of the time, Gnormal crashes 20% of the time, and Halfling crashes 30%
  of the time.
  Sophie is one of Genius Consulting’s ten software testers. Six testers are assigned to Freequen, three to Gnormal, and the last one to Halfling. Sophie’s program crashes when she tests it with a 100-
  variable problem. What is probability that Sophie was assigned to Halfling?