Thursday, February 26, 2015

THAT statement is probably improbable

I saw this article from the fifth person while having mee goreng for lunch. The article talked about how you can increase your wins in investing. All in all, a good piece of article. Sound investing advice. The only complain that I have is that it's not sound mathematically. I'm being anal about this, of course, and it's all because of one word. But let's hear my arguments.

You play a game of tossing a fair coin. If it shows heads, you get $2. If it shows tails, you lose $1.

So what's the expected gains/loss from playing such game? It's 2 x 0.5 - 1 x 0.5 = $0.50. So the article mentioned that if you play one such game, you might not get $0.50, but after playing a large number of games, notably 100 such tosses, the law of averages will kick in and you will earn $50. The article went on to say that "You will always end up profitable after 100 tosses".

Eh...something isn't quite right leh. My mind is attuned to seeing the word 'always'. I get doubts whenever I see that word. Always means 100%. So you mean I will always be profitable after 100 tosses? I've strong issues with that statement that you will always end up profitable after 100 tosses, because it doesn't sound right. Let's test it out.

100 tosses of a coin with a 50% chance of getting heads and 50% chance of getting tails is just a binomial distribution with number of trials as 100 and probability of success, here defined as getting a head, as 0.5. So I let X be the number of heads you get from throwing a fair coin 100 times.

I used an excel spreadsheet to calculate the probability distribution of X. This is a random variable that can only take values between 0 and 100, inclusive. Also, if I have 2 heads, i.e. x = 2, I will lose $94 ($2 x 2 - $1 x 98 = - $94). Below shows a screenshot of the spreadsheet that I did:

This goes on and on until we first see the losses, represented by the strings of negative values, turn positive. That occurs when we have 34 heads and 66 tails, getting us a grand total of $2.

This means that if I get 34 heads and above (i.e. x = 34 to 100), I'll end up having gains after 100 tosses. The probability of getting gains is therefore to sum up all the probabilities from x = 34 all the way to x = 100. That gives me 99.95631%. Quite high, and quite certain, but it's far from 100%.

Conversely, if I get anything from 0 to 33 heads, inclusive, I'll end up having losses after 100 tosses. The probability of losing is thus 0.043686%. Again, it's not certain that we will definitely win. There's a possibility of losing, just that it's highly improbable.

And of course, the expectation of these 100 tosses is still $50 ($0.50 per game x 100 games). I'll be rather careful to say that just because the expectation is $50, you'll always end up being positive after 100 such games. The expectation is just an average of 100 games and we all know what's the perils of using averages. If the average water level in an unmapped ocean is 1 m, does it mean that suddenly, you won't drop into a deep trench of 10 m deep?

Since it's binomial and the distribution is symmetrical, the mode of the distribution, which is the most likely gain/loss you get after 100 toss, will be $50 (corresponding with x = 50), with an attached probability of 7.95892%.

The best that I can say about using expectancy is this: If it's positive, the odds are in your favour. If it's negative, the odds are against you. Probability doesn't mean much to me, having seen my father-in-law winning lotteries almost every month. Yea, so what if the probability is small, as long as there's a chance, no matter how improbable it is, somebody has to win it LOL


Createwealth8888 said...

How many rows of 4D numbers your father-in-law buy per week?


la papillion said...

Hi bro8888,

Not a lot! Once he got feeling he'll buy. He is definitely not those who bought 1000 tickets per month but got 1 ticket in! Lol!

Sillyinvestor said...

Remind me of Graham's Maths theory behind MOS.

Buy MOS and diversify as many as possible, keep buying as MOS widen, mathematically, the odds will be at our favor and we will win eventually

la papillion said...

Hi SI,

That's right! He also cut loss after one year, if I remembered from the book that I read about him. He don't give a damn about the business, just focusing on what is in the financial reports. Basically if you win an amount that is greater than your losses, even if you lose more often, you'll still end up positive.

The cut loss part is an important part of the strategy, not just the part about buying at value! The other important point is that you have to buy a lot of companies for this to work, wouldn't work well on just 10.

Jimmy L said...

not quite convinced about the 100 tosses 'sure win'
sounds like diversification but it will reduce risk but not eliminate risk said...

Hi LP,

Mee Gorenng is not very healthy wor. LoL.

I read the article and I don't feel comfortable with the logic of winning $2 for heads and only losing $1 for tails because no one in the right mind will offer such terms to you in the first place.

I think what the author meant is to have a diversified portfolio where different stocks will do well at different times and that will make up for any under-performing stocks.

And with regards to 4D, the probability is always 50% - either you win or you lose. No? :)

la papillion said...

Hi Jimmy,

You're right :) Can reduce risk but cannot eliminate it haha!

la papillion said...

Hi Derek,

Nothing to eat went down too late so only have mee goreng :)

The game is just an illustration to show that if you win more than you lose, never mind how frequently, you still end up well ahead of others. I'm just picking on little things like 'always' lol!

Undeterred, I went ahead to do a monte carolo simulation on excel. Will publish the results after I set up the blog post lol!

I sibei eng!

darren said...

For things like coin tosses involving probability, it's not possible to say that something will NEVER ever happen (e.g. no losses). True? (Logic is never my strongest suit)

la papillion said...

Hi RT,

Not true. If you throw 100 coins, and you want 101 heads, will it ever happen? I'll say there's absolutely zero chance of it happening LOL

The article said that traders usually do expectancy for their trades...looks like it's not true Hehe