Payoff 1.

Toss 5 coins. You get $1 for each consecutive HT that you get.

Payoff 2.

Toss 5 coins. You get $1 for each consecutive HH that you get.

For example, if you tossed HTHHH, under payoff 1 you will get $1, but under payoff 2 you will get $2.

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TopNewestThis is a trick question, both payoffs have same expected value of profit, which is: $1.

Consider this:

We will count the number of ways HT can occur. We can have:

HT_ _ _ : 8 ways

_ HT _ _ _ : 8 ways

_ _ HT _ : 8 ways

_ _ _ HT : 8 ways

Total number of ways: 32

Similarly for HH, the total number of ways in which it can occur is 32.

What this means is that the sum of the payoffs of HT and HH over the 32 possible outcomes is $32 for both. The payoffs may be distributed in different ways for HT and HH, but their expected value is same.

Mathematically, the expected value is same. But there is one more thing left to do: analyze the probability distribution. We can easily see that payoff 2 might be desirable because it tends to give large profits as compared to payoff 1 for certain events. eg: HHHHH gives $4 in p2 and 0 in p1. HHHHT gives $3 in p2 and $1 in p1.

But the thing is that these events are not very common. Using method of reflection and analyzing the 16 possibilities , we see that there are many events for which p2 gives $0 payoff. There are 13 outcomes for which p2 gives $0 payoff, but only 6 where p1 gives $0 payoff.

So If I'm given this choice, I'd take payoff 1, as then I have more chance of leaving with at least a dollar in my pocket.

Since this is in quantitative finance, should I talk about risk and stuff? I do not know. – Raghav Vaidyanathan · 2 years, 1 month ago

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Indicator Variables, which is essentially what Raghav did (though not phrased in that language).

Yes, the expected payoff for both scenarios is $1. The easiest way to see this is to look atWell, everyone has their own "payoff preference". For example, you stated that you would prefer to "be more likely to leave with at least a dollar in your pocket". As such, this tells me that you are very risk adverse, and that you would prefer the certainty of a positive payoff.

If you want to talk about "risk and stuff", what do you need to consider, and why? – Calvin Lin Staff · 2 years, 1 month ago

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After a bit of thinking, I think I agree with you on the fact that everyone has their own "payoff preference". One can take a risk to win more, or take less risk to win less. In the scenario mentioned here, the aspects of risk are not conspicuous. The amount to be won is not significant, and there is no penalty for losing. Hence, both the payoffs are inherently attractive offers. When we come to real life situations, I think there are many more things that contribute to the risk:

The probability that you will get a payoff.

The probability that you stand to lose money.

The security of the winnings. Money once won shouldn't be taken away from you.

The effects of said decisions on long term/ on your ability to take other decisions.

According to me, every decision of ours is based on weighing in these risk factors against the probable prize. The balance may tip either way, and so may our decisions.

I don't even know if I'm thinking in the right direction... Help me out here @Calvin Lin – Raghav Vaidyanathan · 2 years, 1 month ago

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Hopefully, you will come up with a consistent, logical risk-reward structure. For example, if you would choose payoff 1 over 2 and payoff 3 over 4, but would prefer a combined 2 and 4 over a combined 1 and 3, then it would be very easy for someone to sell you things in part, and make you overpay for them. – Calvin Lin Staff · 2 years, 1 month ago

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Edit: Oh. now I see what you've done; very clever. I must have missed out on my count somewhere. I'll still wait to confirm in the morning. And yes, there does seem to be an advantage to choosing p1, (even though the expected winnings are the same), in the sense that you are more likely to at least win some money. – Brian Charlesworth · 2 years, 1 month ago

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For example,what would you choose between:

Payoff 1: 50% of $0, 50% of $100

Payoff 2: 90% of $1.01, 1% of $4900 – Calvin Lin Staff · 2 years, 1 month ago

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If Payoff 2 was 99% of $0, 1% of $5000, then it gets interesting, (at least for me). Even a such a slight chance of winning $5000 would be hard to pass up, so I would probably go with Payoff 2, unless I absolutely needed that $100 right away.

If Payoff 2 was 99.9% of 0$, 0.1% of$50000 .... hmmmm .... That's a lot of dough, so I'd go with Payoff 2. However, if the options were these last two I've listed, then I might choose the $5000 option.

If Payoff 2 involved a 50% chance of losing $100 and a 50% chance of winning $200, then I would go with Payoff 1, since I'm not comfortable with there being such a good chance of losing a fair bit of money. There is a lot of psychology going on in these choices, and yet the expected winnings is always $50, (which ironically is an amount that would never actually be won in any of these scenarios). – Brian Charlesworth · 2 years, 1 month ago

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So, there are multiple ideas here.

- Is the probability of a positive payoff so important that just because it moves from $0 it would greatly influence your preferences?

- At what point does the potential promise of a huge payoff overweigh the certainty of a small payoff? – Calvin Lin Staff · 2 years, 1 month ago

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– Brian Charlesworth · 2 years, 1 month ago

In general I guess it all depends on the specific values, and further, on each person's financial status and willingness to take on risk. For me, the potential, (1% or better), of a large payout would outweigh any guaranteed amount less than $100, so in this case there is no difference to me between $0 and $100. But if the guaranteed amount were, say, $5000, with a 0.01% all-or-nothing chance of winning $1,000,000, I'd probably take the $5000 and run, (although I would be tempted to take the risk, at least for a moment). If the all-or-noting percentage were 1%, though, then I would find it hard not to take the chance; $5000 is a lot of money, but $1,000,000 is a life-changing amount of money, and a 1% chance is realistic in my eyes given the potential reward. The question I would ask myself is: which will I regret more - not taking the guaranteed money or not taking the risk?Log in to reply

@Calvin Lin sir, Am I right? – Raghav Vaidyanathan · 2 years, 1 month ago

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From a statistical standpoint, I would choose payoff option 1. As the number of H tossed increases, the probability of the consecutively tossing another H decreases. That being said, the HH combination can lead to a higher payoff because HTHTH only results in $2 while HHHHH results in $5. The issue lies in the decreased probability of continuing to toss H. This is a good example related to risk, much like in the decision involved in buying a AAA bond earning 5% vs a B bond earning 13%. – Josh Lester · 9 months, 2 weeks ago

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– Josh Lester · 9 months, 2 weeks ago

To be more specific, payoff 1 would be like the AAA bond and payoff 2 like the B bond. It just depends on what kind of risk is right for you.Log in to reply

Whereas in the bond example that you gave, we are trading expected payoff for certainty.

I also strongly disagree with "As the number of H tossed increases, the probability of the consecutively tossing another H decreases". The coin tosses are independent, and that is a common fallacy that "the proportion of realized events must be equal / close to the calculated probability" – Calvin Lin Staff · 9 months, 2 weeks ago

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I remember reading about an interesting experiment done by a French mathematician regarding payoffs and expected utility from lotteries. Allais Paradox? – Venture Hi · 2 years, 1 month ago

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here, taking a risk, (even if it is only 1%), means the possibility of losing a guaranteed million dollars; I would deeply regret gambling and ending up losing that money, but if I took the million and played 1B just to see what happened and found that I could have won 5 million, I would have just said "Oh well" and still been perfectly happy with my million. But if there is no guaranteed money, then having a chance at 5 million at the expense of a 1% greater chance of winning a million seems worth the risk.

That's interesting. In experiment 1Experiment 1 makes me think of the old saying, "A bird in the hand is worth two in the bush." If there are no "birds in hand", however, then the risk evaluation process is quite different. – Brian Charlesworth · 2 years, 1 month ago

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I think this is a combinatorics problem?! – Agnishom Chattopadhyay · 2 years, 1 month ago

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Notice that I am essentially getting at the risk-reward preference, which is a "finance idea" as opposed to a "combinatorics idea". Yes, combinatorics ideas like expected value is involved, but they don't tell the full story. – Calvin Lin Staff · 2 years, 1 month ago

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There are 2^5 outcoms of 5 coin tosses.

For payoff 1, there are this many ways to toss a HT. There are {HTHTT,HTHTH,HTTHT,HTHHT,THTHT, HHTHT,HTHHH,HHHTH,HHHHT, HTTTT,THTTT,TTHTT,TTTHT,HTTTH,HHTTT, HHTTH} Expected paayyoff 1 = (6/32

$2) + (10/32$1)= $11/16=$22/32For payoff 2, there are this many ways to toss a HH( or HHH,HHHH and HHHHHH). There are {HHHHH, THHHH,HHHHT, THHHT,HTHHH,HHHTH, HHHTT, TTHHH, HHTHH, HHTTT, THHTT, TTHHT, TTTHH, THTHH, THHTH] Expected payoff 2 =(1/32

$4 )+ ( 2/32$3) + ( 5/32$2) + (7/32$1)=$3/4= $27/32Payoff 2 is better, I guess LOL @Calvin Lin

Total possible outcomes of 32 of which there is one combination [TTTTT] which has no payoff. – Venture Hi · 2 years, 1 month ago

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– Calvin Lin Staff · 2 years, 1 month ago

Check your calculations. The expected payoff of both scenarios is $1.Log in to reply

Let Ii be an indicator rv for an event of getting H at ith toss. Let X be a rv which is how many times HT occured. Similarly, let Y be the same but for HH. Then, X = I1

I2 +... +I4I5 and Y = I1(1-I2) +...+I4(1-I5). We need to compare EX and EY. Given EIi = 0.5, we can apply linearity of expectation (even for multiplication since events are independent) and get 1 for both expectations. – Sergei Krestianskov · 1 year, 6 months agoLog in to reply

– Calvin Lin Staff · 1 year, 6 months ago

So, if their expectations are the same, does that mean that you are indifferent about which payoff to take? If so, why?Log in to reply

– Sergei Krestianskov · 1 year, 6 months ago

Yes, because with both payoffs, one can win the same amount of dollars on average.Log in to reply

Payoff 3: Always get $0

Payoff 4: 50% chance of - $1,000,000,000 and 50% chance of $1,000,000,000 – Calvin Lin Staff · 1 year, 6 months ago

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– Sergei Krestianskov · 1 year, 6 months ago

If I don't have a million dollars at all, I will definitely not take payoff 4. I would take payoff 4 only if I was a millionaire. So, psychologically I am not indifferent.Log in to reply

– Calvin Lin Staff · 1 year, 6 months ago

Right, so there are other considerations that come into play. For most people, the certainty of a result is preferred to an uncertain result (though with equal expected value). In general, having a low variance is better (though of course, there are exceptions).Log in to reply