A knight wants to find and kill a dragon who lives in a maze. The knight has a bag of pills, where each pill can either be magical (magical) or not magical (magical). The knight does not know which pills are magical, but does know that the bag contains 5 magical pills and 15 non-magical ones. Thus, the probability of the knight pulling a magical pill from the bag, i.e. P(magical), is 0,25 and the probability of non-magical pill, P(magical), is 0,75.
When fighting the dragon, the knight can either win (win) or not win (—win) the battle. Before encountering the dragon, the knight eats a random pill from the bag. The following then holds:
• Given that the knight ate a magic pill, there is a 70% chance of winning the battle against the dragon, i.e. P(winlmagical) = 0.7 = 7/10. • Given that the knight ate a non-magic pill, there is a 20% chance of winning the battle against the dragon, i.e. P(winl—magical) = 0.2 = 1/5.
Let the two binary random variables M E (magical, —.magical j and W E win, —win) represent the outcome of the pill pulled from the bag and the outcome of the battle, respectively.
Now solve the tasks below. In order to obtain the maximum points, you will have to state your calculations clearly (at least which formula you use and which values goes where), as well as the final probabilities. Also note that some tasks might require you to do a calculation in several steps. Each task is worth 1 point.
Task a. What are the marginal probabilities of W, i.e. what is P(win) and P(win)?
Task b. Calculate the conditional probabilities P(—win I magical) and P(win I —.magical).
Task c. What is the probability that the knight ate a magic pill, given that the battle was won?
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