Question

There are three blocks, floating in a sea of lava. Label the blocks 1,2,3, from left to right. Mark the Kangaroo, a video game character, is standing on block 1 . To reach safety, he must get to block 3 . He can't jump directly from block 1 to block 3 ; his only hope is to jump from block 1 to block 2 , then jump from block 2 to block 3 . Each time he jumps, he has probability 1/2 of success and probability 1/2 of "dying" by falling into the lava. If he "dies", he starts again at block 1. Let J be the total number of jumps that Mark will make in order to get to block 3 .

(a) Find E(J), using the Markov property and conditional expectation.
(b) Explain how this problem relates to a coin tossing problem

Answer 1

To find E(J), we can use the Markov property and conditional expectation. We break down E(J) into two cases: if Mark jumps successfully from block 1 to block 2, or if Mark falls into the lava. Solving the equation, we find that E(J) = 2. This problem relates to a coin tossing problem because each jump Mark makes has a 1/2 probability of success and a 1/2 probability of failure, similar to flipping a fair coin.

Step-by-step explanation:

To find E(J), we can use the Markov property and conditional expectation. Let's denote E(J) as the expected number of jumps to reach block 3 starting from block 1. We can break down E(J) into two cases: if Mark jumps successfully from block 1 to block 2, or if Mark falls into the lava.

Case 1: If Mark jumps successfully from block 1 to block 2, he will need an additional E(J) jumps to reach block 3. The probability of successfully reaching block 2 is 1/2. Therefore, the expected number of jumps in this case is (1/2) * E(J).

Case 2: If Mark falls into the lava, he will need to start again from block 1 and make an additional E(J) jumps. The probability of falling into the lava is also 1/2. Therefore, the expected number of jumps in this case is (1/2) * E(J).

Considering both cases, we can write the equation: E(J) = (1/2) * E(J) + (1/2) * E(J) + 1. Solving this equation, we find that E(J) = 2.

(b) This problem relates to a coin tossing problem because each jump Mark makes has a 1/2 probability of success and a 1/2 probability of failure, similar to flipping a fair coin.