The entire point of this post was to ask whether there was a general approach to handling gambling problems, and everyone here is instead nitpicking the OPs example about horse racing, which actually has nothing wrong with it. The house offers a payout on possible outcomes, and one has determined by some means what the probabilities of those outcomes are, which may or not be accurately reflected by the payout. It's not rocket science.

The mathematics of it is what the OP is asking about, not some treatise on betting markets.

If the math of my response is incorrect, then please illuminate me.

Well, i have no problem with someone being opinionated if they are correct, and the net sum of votes is not always a reflection of that. Most people have TLDR attitude, and it seems one of my downvotes came from the OP herself lol, the very person who is not clear what the correct answer is.

I think you are right about that. i apologize for being rude. There is a way to take issue and be respectful, and that hasn't been me in this conversation.

@RodrigodeAzevedo Look that was the question given to my friend in an interview. I don't think there is a right answer. I am just interested in the approach on how to reason about this! You don't want to invest serious time in reading the answers and yet you are investing serious time arguing.

@RyRytheFlyGuy I did not down vote you, but I did not accept it either. There isn't a correct answer. I am skeptical about the first part because for a 7:2 odds, if pink wins, I get 7 + 2 = 9 dollars or 9/2 odds. Just think about the inverse, for example, if you have 2:7, I would win 2 + 7 also but now I can lose 7 instead of 2. Wining only 2 and not get your 7 back isn't winning exactly.

@Sarah you do not appear to understand what odds means. $7:2$ odds implies that for $\$2$ risked, you can win $\$7$ more. So if you bet $\$2$ and win, then you get your $\$2$ dollars back plus $\$7$ more. $2:7$ odds implies that for $\$7$ risked, you can win $\$2$ more. So if you bet $\$7$ and win, then you get your $\$2$ dollars back plus $\$7$ more.

@RyRytheFlyGuy Yes that what I mean

ok. I think part of the issue here is that you have posted multiple questions about several topics in one post. You want a general approach to betting problems... then you ask about trading stocks... then you ask about a very specific example in horse betting. I would split these up and ask specific questions about each.

@RodrigodeAzevedo Why don't you lecture me then if it is trivial? You spent your whole life good will hunting, but the only good that will come is when you start helping!

Imagine the horse race is performed $1000$ times and every time you bet on the purple horse. In approx. $1000p$ races, purple wins and you make $1000p\$\frac{9}{2}=1000p\$4.5=\$4500p$; in approx. $1000q$ races, blue wins and you lose $1000q\$1=\$1000q$; and in approx. $1000(1−p−q)$ races, black wins and you lose $1000(1−p−q)\$1=\$1000(1−p−q)$

So your reward over $1000$ races is approx $\$4500p−\$1000q−\$1000(1−p−q).$ That is your expected reward over $1000$ races. You have to take the reward from all bets you expect to win and subtract from it the cost of all bets you expect to lose

You will have to explain why you believe $E[purple] = \frac{9}{2}p$

Hi Ry

hey!

Look first of all I really appreciate you trying to help

no probelm. i also got annoyed with others earlier

so many thanks. I also saved your answer locally even though it was deleted. I am just trying to understand this

very nit picky and not trying to be helpful

okay good

so where do you run into confusion?

I think the expected value is just 9/2 x p because I am only familiar with the definition of expected value, that is just multiply each value of the random variable by its probability?

Okay, so let's define a random variable that called "PINK" such that PINK = K if and only if K is that amount of profit you get after betting on the pink horse

Why is it $3.5 in BetPurple=3.5p−1q−1(1−p−q)?

awesome

actually

so you bet on the pink horse... what are the possible outcomes? either the pink horse wins, the blue horse wins, or the brown horse wins, correct?

yeah

How much profit do you make if you bet pink and... pink horse wins? blue horse wins? brown horse wins?

hmmm 9/2 for pink and -1 for blue and brown?

correct! so then your random variable PINK can assume two values: either 9/2 or -1

yes

okay great... what is the probability that pink wins? and what is the probability that blue or brown wins?

okok I think I understand

yayyy

how do I maximize max(BetPurple,BetBlue,BetBlack)

you have the right idea about expectation. simply multiply each value of the random variable by its corresponding probability and add them all up. You do the same here with PINK

okay so to answer your question about maximization...

Also why do I have to use the kelly criterion to pick my initial bet? I am also looking at this post with similar question math.stackexchange.com/questions/3777086/…

the calculation we just did with PINK... you need to do the same for random variables BLUE and BROWN. When finished, you have three expectations, one for PINK, BLUE, BROWN. Then, in order to determine which horse to bet on, you simply look at which expectation is the largest. That is what I mean by max(E[PINK], E[BLUE], E[BROWN])

you are just finding which expectation happens to be the largest or "max"

so If I know p and q, I can compute the expected value and then I know which to bet on? So now choosing the size of my bet, why don't I just bet all (or a portion) with my money on the horse with the highest expected value? This will maximise my expected return?

sorry yes

I meant the size of my bet

okay cool you tracking great so far

The reason you do not bet all your money at once is because there is uncertainty as to which horse will win and you could potentially lose everything in one race

technically betting all of your money at once will maximize the expected return as you are saying... but it does not maximize the rate of growth of your wealth because your rate of growth could potentially go to zero after one race

ahhh I see, I didn't think about the rate of growth

so this is where the kelly criterion comes in... it is intended to maximize the rate of growth of your wealth while accounting for the uncertainty that is involved in each bet

wooo you got it

Ahhh I think I understand this now. I still need to spent some more time understanding the kelly criterion

I really appreciate your help ry!

yeah i included some links to papers about it in my post. the derivation of the formula might be complicated but the end result is easy to follow

yes I saved them already :)

yeah no problem

I downloaded your answer just in case

are you in university?

yes

awesome. what has sparked this interest of yours in gambling?

bio said netherlands but i am in Aus

oh cool i want to go to Australia someday

Idk I just think its very interesting and useful?

My friends are also interested in stats, I got this question from my friend

yeah definitely. i am intrigued by the whole idea of using math to wrestle with uncertainty in general.

i am studying machine learning here in the states

I am doing CS with a minor in applied maths

GREAT combo

but I think you are better XD

are you gettn bachelor?

yeah

u?

okay i am doing Masters... actually masters #2. so if i seem like have good handle on this, that's why. time plus effort is everything. the fact you are online asking questions and digging in means you have nothing but great things ahead

yeah I knew it, you just way better than me. Haha thanks It just really bother me not knowing how to do something.

haha wow i am the same way

i get really annoyed by not understanding something

do u have an undergrad in maths?

no lol i actually obtained a Bachelor of Social Work of all things

oh really? You seem to be really good at maths/stats

oh well, there is plenty of rigour in machine learning

yeah i've had a strong interest in math, programming, logic, AI, etc for a while now. I went back to school for Master in Biostatistics and then back again for CS/machine learning (what i'm studying currently)

damn wow!

yeah the program i am in now is brutal

but brutal in a good way

yeah a lot of stats

for sure

what do you want to do with CS?

machine learning too

oh awesome!

Also ry I think I found a a very similiar question here math.stackexchange.com/questions/4522742/…

i have found myself really intrigued by the philosophical/theoretical aspect of ML

Oh cool I see I haven't reached that far yet in my degree. So far its just here is an algorithm, just apply it to some dataset. Not yet into the theorical

okay i will look over that link

an important thing to keep in mind when sizing bets is (1) are there minimum bet amounts? (2) are there maximum bet amounts? (3) how many bets are allowed to make?

the kelly criterion assumes no limit on any of these

and additional constraints can change things

anyway, i have to get back to a project. nice chatting with ya. good luck!

No worries, ry! Thanks so much for helping me! I wish I could have accepted your answer! Many thanks again! @RyRytheFlyGuy

no problem. i look forward to seeing you on TV one day for all your successes