RyRy the Fly Guy

@Dmitry if you are going to edit and update your OP, then please explicitly indicate that you are doing so by saying "EDIT (date)." By changing the OP without notice, you make all existing comments and answers appear increasingly irrelevant when they were not in the first place.

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

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

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 $1000(1−p−q)\$1=\$1000(1−p−q)$

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.

@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.

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.

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.

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

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

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.

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

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

and additional constraints can change things

the kelly criterion assumes no limit on any of these

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?

okay i will look over that link

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

oh awesome!

what do you want to do with CS?

for sure

but brutal in a good way

yeah the program i am in now is brutal

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)

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

i get really annoyed by not understanding something

haha wow i am the same way

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

are you gettn bachelor?

GREAT combo

i am studying machine learning here in the states

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

oh cool i want to go to Australia someday

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

are you in university?

yeah no problem

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

wooo you got it

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

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

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

okay cool you tracking great so far

The Kelly Criterion is not used to determine which horse to bet on. The calculations we just went over will tell you which horse to bet on. The kelly criterion simply tells you how much of your bankroll you need to risk

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

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])

okay so to answer your question about maximization...

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

yayyy

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

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