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## Homework Statement

(a) Suppose we flip a fair coin until two Tails in a row come up. What is the expected number, NTT, of flips we perform? Hint: Let D be the tree diagram for this process. Explain why D = H · D + T · (H · D + T). Use the Law of Total Expectation

(b) Suppose we flip a fair coin until a Tail immediately followed by a Head come up. What is the expected number, NTH, of flips we perform?

(c) Suppose we now play a game: flip a fair coin until either TT or TH first occurs. You win if TT comes up first, lose if TH comes up first. Since TT takes 50% longer on average to turn up, your opponent agrees that he has the advantage. So you tell him you’re willing to play if you pay him $5 when he wins, but he merely pays you a 20% premium, that is, $6, when you win. If you do this, you’re sneakily taking advantage of your opponent’s untrained intuition, since you’ve gotten him to agree to unfair odds. What is your expected profit per game?

## Homework Equations

Law of Total Expectation

##E[R] = \sum_i E[R | A_i] Pr{A_i }##

## The Attempt at a Solution

I have the solutions, and am confused over part a.

Solution a):

N

_{TT}= 6.

From D and Total Expectation:

## N_{TT} = \frac{1}{2} * [1+N_{TT} ] + \frac{1}{2} * (1 + \frac{1}{2} * [1+N_{TT} ] + \frac{1}{2} *1 ) ##

So, it's to my understanding that to answer these kinds of questions, the tree is D, while H/T are the probabilities of heads/tails, which are each 1/2. If the first toss is heads, that doesn't further you towards your goal at all, and basically leads you to a sub-tree that is identical to D. That is where the H*D comes from in the Hint. If you get a Tails on first toss, for your subsequent toss you have a the possibility of tossing a heads and starting all over (H*D), or a chance of getting a tails which ends the game (T) for a combination of T( H*D + T). So the hint's equation seems reasonable to me because you can combine the expressions for if you tossed heads first and if you tossed tails first to get : D= H*D + T*(H*D + T)

My issue is that I am not clear what to do with that equality. It seems like you want to replace the instances of D on the right hand side of the equation with [1+ N

_{TT}] because anytime D appears on the RHS, that means you have rolled a heads and are back to the beginning situation and you will need the N

_{TT}number of turns to end the game from this point. The "1+" before the N

_{TT}is to indicate that you wasted a turn because of you tossed a heads.

The LHS "D" is replaced with N

_{TT}to indicate the number of turns.

Is this reasonable so far? My issue with this problem is the addition of the 1 inside of the parenthesis (). The second "1" to appear in this equation, left to right. It is just after the left parenthesis "("

## N_{TT} = \frac{1}{2} * [1+N_{TT} ] + \frac{1}{2} * (1 + \frac{1}{2} * [1+N_{TT} ] + \frac{1}{2} *1 ) ##

I do not understand where that 1 came from. It is not being added to an N

_{TT}term, so I don't think my prior logic applies. I don't see it any any other similar problems, but it appears to be critical to this problem.