1-a Please only if you have experience with Stochastic Operations Research or En

Question

1-a Please only if you have experience with  Stochastic Operations Research or Engr Statistics - Probability

answer this question, please leave it if you not fully sure, I am not on a rush "I can wait for 24 hours"

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Let's toss a coin continuously and independently (a Bernoulli process In general, the probability of coming up head in a toss is p. If the second head happened at the nth toss, what is the chance that the first head happened at the ith toss? Note that i could be 1, or 2, till to n 1. Discuss why the final solution of this problem has nothing to do with either p or i.

Explanation / Answer

Solution:

first head could be at at 1 or 2 ...(n-1)

A = P(i) = the chance that the first head happened at the ith toss

B = event that second hand is at nth toss

{T,T,T,T,T....T,H,T,T...T.T} here H is at ith toss.

P(A|B) = P(AB)/P(B) =(1-p)^(n-2) * p^2/((n-1)*(1-p)^(n-2) * p^2)

= 1/ (n-1)

since

P(i =1) =P(i =2) = ... P(i = n-1)

so required probability = 1/(N-1)

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