4. (5 points) Islands 0,1,2 and 3 are next to one another. You are on island zer

Question

4. (5 points) Islands 0,1,2 and 3 are next to one another. You are on island zero and you want to get to island 3. Between islands i and i + 1 (i = 0, 1, 2), there are two bridges: one is a regular bridge but the other one is broken and will collapse if you step on it. Once a bridge collapses, you fall into the river and have to swim back to island 0. The first time you arrive at island i (i = 0, 1, 2), you choose randomly (with probability 1/2- 1/2) which bridge to take to get to island i +1. Let X be the number of times you completely cross a bridge (i.e.: broken bridges not counted, but the good bridges counted as many times as you crossed them). Find the p.m.f. of X

Explanation / Answer

Solution:

There are 8 ways on one can travel from island 0 to island 3.

Let unsuccessful bridge be U and successful be S. Now, there are 8 possibilities.

{UUU, UUS, USU, USS, SUU, SUS, SSU, SSS}

All of them have probability ½ * ½ * ½ = 1/8

Each time guy goes ones on unsuccessful bridge, the next time he will go to the correct bridge. So the number of x for each event are as follows:

UUU = 6 steps

UUS = 4 steps

USU = 5 steps

USS = 3 steps

SUU = 6 steps

SUS = 4 steps

SSU = 5 steps

SSS = 3 steps

X PMF

3         2/8 = 1/4

4         2/8 = 1/4

5         2/8 = 1/4

6         2/8 = 1/4

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