Choosing TT vs TH problem

[tuanl]

Suppose you roll a fair coin 5 times. You can either be paid the number of consecutive tails "TT" or the number of "TH"'s. Which one would you choose and why? For example, the sequence "TTHTT" would count 1 instance of "TH" and 2 instances of "TT."

Generalize your result for n flips. Is the result different now?

 

[IlyaKEightSix]

TT. Think about it. If you choose TH, then at most, you can have 2 sequences in your 5 flips. But if you choose TT, consider the case of getting lucky with five tails in a row: TTTTT.

How many instances of TT are there? Not 2, but 4. The reason for that is when you roll TH, then the next combination can't possibly be another TH. However, when you have TT, then the second T can be the first T in the next sequence. This basically gives you twice the chance.

Now if it was TH AND HT vs. TT (or HH for that matter), we'd have another discussion.

 

[tuanl]

TT. Think about it. If you choose TH, then at most, you can have 2 sequences in your 5 flips. But if you choose TT, consider the case of getting lucky with five tails in a row: TTTTT.

How many instances of TT are there? Not 2, but 4. The reason for that is when you roll TH, then the next combination can't possibly be another TH. However, when you have TT, then the second T can be the first T in the next sequence. This basically gives you twice the chance.

Now if it was TH AND HT vs. TT (or HH for that matter), we'd have another discussion.

I think "TH" is the answer based on calculating the P(getting at least one TH out of 5 tosses)(= 26/32=13/16)>P(getting at least one TT out of 5 tosses)((=19/32)). Notice that expected payoff of TT=the expected payoff of TH=1. So if you're less risk averse, we can choose TT and hope to get lucky with TTTTT.

I haven't worked on the general case yet, but I'm pretty sure the same result still holds(we should still choose TH).

 

[amanda.jayne]

see

http://academic.cengage.com/resource_uploads/downloads/1133105068_339606.pdf

read pages 1157 onward. Add in weights.

(google is your friend)

 

[pruse]

It ain't about googling, it's about sharing and discussing; I'm sure that's why tuanl posts his problems here.

Use indicator variables: For each of the first $n-1$ tosses, out of $n$, let $X_1=1$ if that toss and the next are TH, 0 otherwise. Then the expected number of TH's is equal to $E(X_1+\cdots+X_{n-1})=(n-1)E(X_1)=(n-1)P(X_1=1)=\frac{n-1}{4}$. The same argument shows that the number of TT's is expected to be the same. So it's the variance that will decide.

Now for the variance. \(Var(X_1+\cdots+X_{n-1})=E((X_1+\cdots+X_{n-1})^2)-\large(\frac{n-1}{4}\right)^2\). $E((X_1+\cdots+X_{n-1})^2)=(n-1)E(X_1^2)+2\sum_{i<j}E(X_iX_j)$. Clearly $(n-1)E(X_1^2)$ will be the same in both the TT and the TH cases. We need to see how the $\sum E(X_iX_j)$ part differs.

1) TH case. When $j-1=1$, $X_iX_j$ clearly must equal 0. In all other cases, disjoint pairs of tosses are involved and these are independent, so $E(X_iX_j)=P(X_iX_j=1)=P(X_i=1,X_j=1)=P(X_i=1)P(X_j=1)=\frac{1}{4}\cdot\frac{1}{4}=\frac{1}{16}$.

2) TT case. This is clearly larger as the disjoint pairs of tosses give the same $E(X_iX_j)$ as in the TT case, and the overlapping pairs give a positive $E(X_iX_j)$, whereas in the TH case these are 0.

So it's better to pick TH.

 

[IlyaKEightSix]

I think "TH" is the answer based on calculating the P(getting at least one TH out of 5 tosses)(= 26/32=13/16)>P(getting at least one TT out of 5 tosses)((=19/32)). Notice that expected payoff of TT=the expected payoff of TH=1. So if you're less risk averse, we can choose TT and hope to get lucky with TTTTT.

I haven't worked on the general case yet, but I'm pretty sure the same result still holds(we should still choose TH).

How is P(TT) < P(TH)? P(T)=50%. P(H)=50%. P(T)*P(T)=25%. P(T)*P(H)=25%.

Is there something biased about this coin?

 

[tuanl]

How is P(TT) < P(TH)? P(T)=50%. P(H)=50%. P(T)*P(T)=25%. P(T)*P(H)=25%.

Is there something biased about this coin?

I said P(getting AT LEAST 1 TT)<P(getting AT LEAST 1 TH). And no, the coin is definitely fair.

 

[IlyaKEightSix]

I said P(getting AT LEAST 1 TT)<P(getting AT LEAST 1 TH). And no, the coin is definitely fair.

Well, what about the probability of getting 2 TTs vs. 2 THs? With 2 TTs, you can have TTTHH HTTTH, HHTTT, TTHTT, HTTTT, and so forth...

With TH, you can only get two of them at MOST on five flips.

 

[tuanl]

It ain't about googling, it's about sharing and discussing; I'm sure that's why tuanl posts his problems here.

Use indicator variables: For each of the first $n-1$ tosses, out of $n$, let $X_1=1$ if that toss and the next are TH, 0 otherwise. Then the expected number of TH's is equal to $E(X_1+\cdots+X_{n-1})=(n-1)E(X_1)=(n-1)P(X_1=1)=\frac{n-1}{4}$. The same argument shows that the number of TT's is expected to be the same. So it's the variance that will decide.

Now for the variance. \(Var(X_1+\cdots+X_{n-1})=E((X_1+\cdots+X_{n-1})^2)-\large(\frac{n-1}{4}\right)^2\). $E((X_1+\cdots+X_{n-1})^2)=(n-1)E(X_1^2)+2\sum_{i<j}E(X_iX_j)$. Clearly $(n-1)E(X_1^2)$ will be the same in both the TT and the TH cases. We need to see how the $\sum E(X_iX_j)$ part differs.

1) TH case. When $j-1=1$, $X_iX_j$ clearly must equal 0. In all other cases, disjoint pairs of tosses are involved and these are independent, so $E(X_iX_j)=P(X_iX_j=1)=P(X_i=1,X_j=1)=P(X_i=1)P(X_j=1)=\frac{1}{4}\cdot\frac{1}{4}=\frac{1}{16}$.

2) TT case. This is clearly larger as the disjoint pairs of tosses give the same $E(X_iX_j)$ as in the TT case, and the overlapping pairs give a positive $E(X_iX_j)$, whereas in the TH case these are 0.

So it's better to pick TH.

Nice solution, but the computation of the variance really bothers me. I know it's a common technique to compare the variance of two events to make a decision when the expected values of these events are equal. But I wonder if my reasoning above also works in this case, since we clearly see that probability of not getting any money when choosing TT is greater than that of TH.

 

[tuanl]

Well, what about the probability of getting 2 TTs vs. 2 THs? With 2 TTs, you can have TTTHH HTTTH, HHTTT, TTHTT, HTTTT, and so forth...

With TH, you can only get two of them at MOST on five flips.

But HTTTT counts as 3 TTs. In fact, probability of getting exactly 2 THs=probability of getting exactly 2 TTs. But we have to calculate the total probability of all the possibilities that can happen with TH and TT. Basically, choosing TH always give you a better chance to get some money than choosing TT. Notice that this reasoning works, in my opinion, since the expected value of TT=expected value of TH. The proof for this, for general value of n, was presented above by pruse above.

 

[pruse]

Nice solution, but the computation of the variance really bothers me. I know it's a common technique to compare the variance of two events to make a decision when the expected values of these events are equal. But I wonder if my reasoning above also works in this case, since we clearly see that probability of not getting any money when choosing TT is greater than that of TH.

The probability of not getting any money doesn't tell the whole story, as it's possible for the cases that do make money to make disproportionately larger amounts.

Your expected winnings, as well as the volatility of those winnings, are what matter. Now, between two alternatives with the same expected value, the one with the smaller volatility is generally preferred since your confidence interval is smaller.

 

[IlyaKEightSix]

Hmmm, I did a dirty computational solution in R and it seems no matter what happens, when you add up the sum over all the combinations, the sums of both THs and TTs come out identical, but the variance of the THs are lower.

Seems I'm mistaken.

 

[pruse]

Yeah, that's already been established.

 

[Kangxi]

Hey, my answer for 5 tosses is as follows:
1 TT: 3+2+2+3 = 10
2 TT: 1+(2+1+2) = 6
3 TT: 1+1 = 2
4 TT: 1
So the expected money received if choosing TT will be (10*1 + 6*2+2*3+4*1)/(2^5)=1

1 TH: (2^3-4) + (2^3-2)+(2^3-2)+(2^3-4) = 20
2 TH: 2+2+2 = 6
So the expected money received if choosing TH will be (20*1 + 6*2)/(2^5)=1

Therefore, no matter what I choose, I will be paid $1 in expectation.

 

[tuanl]

Hey, my answer for 5 tosses is as follows:
1 TT: 3+2+2+3 = 10
2 TT: 1+(2+1+2) = 6
3 TT: 1+1 = 2
4 TT: 1
So the expected money received if choosing TT will be (10*1 + 6*2+2*3+4*1)/(2^5)=1

1 TH: (2^3-4) + (2^3-2)+(2^3-2)+(2^3-4) = 20
2 TH: 2+2+2 = 6
So the expected money received if choosing TH will be (20*1 + 6*2)/(2^5)=1

Therefore, no matter what I choose, I will be paid $1 in expectation.

But choosing TH gives you smaller variance than TT, which means TH is less volatile than TT. So you should choose TH

 

[Kangxi]

But choosing TH gives you smaller variance than TT, which means TH is less volatile than TT. So you should choose TH

Yes, I think so.