Question

John is lying on the sidewalk after robbing a bank, in pain and mulling over how to quantify the uncertainty of his survival, when Dirty Harry walks over. Dirty Harry pulls out his 44 Magnum and puts two bullets opposite each other in the six slots in the cylinder (e.g., if you number them 1 .. 6 clockwise, he puts them in 1 and 4), spins the cylinder randomly, and, saying "The question is, are you feeling lucky, probabalistically speaking, computer science punk?" points it at John head and pulls the trigger.... "CLICK!" goes the gun (no bullet) and Dirty Harry smiles... "How about that .... Let's see if this gun is memory-less!" Without spinning the cylinder again, he points the gun at Wayne's head and pulls the trigger again.

(a) What is the probability that (at least in my dream) John is hit?

(b) Now, suppose that when Dirty Harry put the bullets in the gun, he put them right next to each other (e.g., in slots 1 and 2). He spins it as usual. What is the probability in this case John is hit?

(c) Suppose Dirty Harry puts the bullets in two random positions in the cylinder and we don't have any idea where they are. He spins it as usual. Now what is the probability that John will be hit?

Answer 1

SOLUTION:-

Part (a)

Since the two bullets are loaded into two opposite slots and the first shot did not have the bullet, the slot opposite to that slot also cannot have any bullet. Thus, two loaded slots are necessarily 2 among the remaining 4 slots. Thus, the probability the next shot will have bullet is 2/4 = 0.5 ANSWER

Part (b)

Here the loaded slots are adjacent to each other. Let, for the sake of convenience and without loss of generality, the slot from where the first slot went be numbered 1. Since that shot was blank, the two bullets are in Slots2&3 or Slots3&4 or Slots4&5 or Slots5&6. Out of these 4 cases, only in the first case there is a bullet in Slot 2. So, the probability the next shot will have bullet is 1/4 = 0.25 ANSWER

Part (c)

Given that the loading of bullet is random and the first shot did not have the bullet, there are two loaded slots out of the remaining 5 slots. So, the probability the next shot will have bullet is 2/5 = 0.4 ANSWER