Can you use a difference equation and can you show your steps? Consider the five

Question

Can you use a difference equation and can you show your steps?

Consider the five symbols A,B,C,D.E. A string of letters is written using these five in the following way. The letter A is written first. No letter is written twice in a row. After a letter is written, the next letter written is equally likely to be any of the other four. Formulate a difference equation to determine the probability pn that the letter B is the nth letter written. Use the law of total probability. Specify the initial conditions.

Explanation / Answer

There are 5 position to fill. The first position is filled by A, the next four are empty. Now B can be in either 2,3,4 or 5th position. Let us call probability in 2nd position is P2, 3rd is P3, 4th in P4 and 5th is P5.

As B has to be in any of the 4 positions, P2 + P3 + P4 + P5 = 1.

As B is eqally likely to be in any of the four positions, P2,P3,P4,P5 are all equal. So P2,P3,P4,P5 will all be 1/4. So the probability of B is the nth letter position is 1/4.

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