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A coin tossing challenge!

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A fair coin is tossed repeatedly.
1. What is the expected number of tosses before HT shows up for the first time?
2. what is the expected number of tosses before TT shows up for the first time?
3. What is the probability that HT shows up before TT?

Thank you for your help.

Oct 4, 2018

#1
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$$\text{Consider a win on the nth roll.}\\ \text{This roll sequence looks like }\\ \underset{n_T}{\underbrace{TTT\dots T}}\underset{n_H}{\underbrace{HHH\dots H}}T\\ 0\leq n_T \leq n-2,~1\leq n_H \leq n-1,~n_T+n_H +1 = n$$

$$\text{Thus there are }n-1 \text{ ways to accomplish a win on the nth roll}\\ \text{Each sequence has probability }p_n = 2^n \text{thus the probability of a win on the nth roll is given by}\\ P[n] = \dfrac{n-1}{2^n},~~n\in \mathbb{N},~2 \leq n$$

$$E[N] = \sum \limits_{n=2}^\infty~\dfrac{n(n-1)}{2^n}=6-2= 4$$

You can apply the same idea to solve 2 and 3

Oct 4, 2018