Can you explain what you\'re doing and what the initial conditions are? Consider

Question

Can you explain what you're doing and what the initial conditions are?

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

A is fixed as the first letter.

Now as no letter is written twice and after A is written all letters are equally likely to follow hemce we have

Now total number of ways to fix the rest of the 4 letters is 4! = 24 = total number of outcomes

there are 4 ways to fix the first 2 letters

A,B

A,C

A,D

A,E

For A,B as the first 2 letters there are 3!=6 ways to arrange the rest of the 3 letters

For A,C are the first 2 letters, there are 3!=6 ways to arrange the rest of the 3 letters ,out of which ,B would come twice in 3rd place (A,,C,B,E,D)(A,C,B,D,E) twice in 4th place (A,C,D,B,E)(A,C,E,B,D) and twice in 5th place (A,C,D,E,B)(A,C,E,D,B)

Like wise For A,D are the first 2 letters, there are 3!=6 ways to arrange the rest of the 3 letters ,out of which ,B would come twice in 3rd place twice in 4th place and twice in 5th place.

Like wise For A,E are the first 2 letters, there are 3!=6 ways to arrange the rest of the 3 letters ,out of which ,B would come twice in 3rd place twice in 4th place and twice in 5th place.

P2(Prob of B coming at 2nd place ) =6/24

P2(Prob of B coming at 3rd place ) =6/24

P2(Prob of B coming at 4th place ) =6/24

P2(Prob of B coming at 5th place ) =6/24

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