A "hardcore" mathematics question

At initial inspection, I thought your Excel program was right on, nice work! (as I look at it at this moment).

After coffee, I tweaked the program and ran some numbers for you to compare as you asked.

Same rules, but different win %, this is what I came up with:

winning trade % ...........chance to hit -10 units in 30 trades at RRR 1:1

50% .......... ................4.18%
40% .......... ................25%
30% .......... ................64%

I hope that helps... fun stuff guys, thanks for the thoughts and contributions.

I've skimmed through this thread a couple of times since yesterday, but I haven't been clear on what the OP was asking until now (I think). If my understanding of the question is correct then there IS an exact solution to any such problem, but the computation might be a bit tedious.

The question isn't "what's the probability of there being >= 10 losses in the sequence?" but rather "what's the probability that AT SOME POINT in the sequence the number of losses has exceeded the number of wins by 10 or more?"

If the number of trials is known and finite, then we can find an exact answer to this question without a Monte Carlo approach. We can do it with a programming approach (I'm with Craig...nice to see some old BASIC code in here; w00t!! ).

Unfortunately I'm heading out right now, but I'll be back in 3 or 4 hours to provide a proposed solution in detail.

I agree, there is an exact solution because it is a finite number of trades (30). A path to the approach is buried in Craig's link previously, the information on random walk, I believe.

Or-

Using the 60/40 win/loss %:

It is basically the sum of all permutations of trade sequences (in a 30 trade sequence) that produce -10 units or more... divided by the sum of all possible permutations of a 30 trade sequence. I think. That would yield the probability

Unfortunately, this kind of math is not my strong point.

Back in a few hours or more visiting relatives, followed by work.

That would be my thoughts, starting with 10 consecutive opening losses through 11 consecutive opening wins or 11 total wins in the 30 set, every permutation divided by the total possible, however this figure needs to be weighted for the 60/40 as the probability for any losing or winning sequence would be proportional to the overall R:R, if I'm thinking correctly

First of all, I'm still unclear on one point regarding what constitutes a "drawdown of 10 units" in the given example. On the one hand it could simply mean that at some point in the sequence the total loss equals 10 units. On the other hand it could refer to the drawdown from the high water mark of the sequence.

Under the first interpretation, we could have a sequence with 11 consecutive wins followed by 19 consecutive losses, and this would NOT trip the drawdown threshold because the total loss over the entire sequence would never exceed 8 units.

Under the second interpretation, the sequence I described above would trigger the drawdown threshold at the 21st trade, because that would represent a 10 unit loss from the high water mark equity of +11.

The good news is that I don't really need to know which interpretation of the "rules" is correct, because the general approach I have in mind will accommodate either one. I'll describe the approach in the next post.

Also note that "consecutiveness" has little if any bearing on the problem. We could have a sequence like this:

WLLWLLWLLWLLWLLWLL.... that repeats for the entire 30 trade sequence. At trade #15, this triggers the 10 unit drawdown threshold.

This is just the binomial distribution ie probabilty applied to 2 outcomes: win or loss?

very nice!

This is my best guesses by using pascal triangle. My logic is similar to the working principle of one touch option, where you will get paid if the price ever touch on your predetermined level within the predetermined period. In this case the predetermined level is -10units and predetermined period is 30 trades.

winning trade % ...........Discrete probability to 'touch' -10 units within 30 trades at RRR 1:1
70% ........... 0.02%
60% ........... 0.59%
50% ........... 7.08%
40% ........... 34.09%
30% ........... 76.52%
20% ........... 97.91%

This will be my last try. Most likely i were wrong again because my numbers went too far from Intu's simulated numbers . I think the logic maybe much more complex than my imagination and beyond my capability.

I look forward for capitalist88's solution.

The approach I have in mind is basically a brute force computational one which involves:

1. checking every permutation of the trade sequence
2. determining the probability of that permutation occurring
3. determining if that permutation hits the drawdown threshold
4. adding up the probabilities of all the permutations that hit the threshold

Writing a program in BASIC or any other scripting language to do this would be fairly straightforward. First determine the number of possible permutations, N. Then initialize a probability (final answer) variable with a value of 0.

Then construct a loop that goes from 0 to N-1. This numbers the permutations in such a way that a string of all zeros represents 30 losses in a row, up to the string of all ones indicating a sequence of 30 wins.

Inside that loop, find the binary string that represents the current permutation (do this with a subroutine that uses the modulo of repeated divisions by 2 in order to build the string from right to left). Initialize a variable for the probability of this permutation occurring and set it equal to 1.

Then construct an inner loop that looks at each position in the binary string. If it's a 1 (win) then multiply the probability variable by 0.6, and if it's a 0 (loss) then multiply by 0.4. The final value of the probability variable is the probability that this particular sequence occurs. An alternate method would be to calculate probability as 0.6 to the power of the number of wins times 0.4 to the power of the number of losses. Either way gives the same answer.

Jump out of that inner loop and jump down into another one which checks to see if the drawdown threshold is hit for this permutation. The structure of this part depends on the actual "rules" for what actually constitutes a drawdown of 10 units, but in any case it should be straightforward to code. If it turns out that this particular permutation DOES hit the drawdown threshold, then add the probability found in the previous inner loop to the "final answer" probability variable.

Jump out of both loops and display the final answer.

In the next post I'll give a concrete example of this using MUCH smaller numbers than the OP's original example.

yea intu did it right its about I got about 0.027% chance after 10,000 iterations. Just do a montecarlo simulation in excel. If you want to get close to the exact answer just to a million iterations.

A quick word on the difference between permutations and combinations. Combinations don't take into account the order in which the choices are arranged, but permutations DO take order into account. We're dealing with permutations because this problem DOES involve the order in which the wins and losses take place. 10 losses followed by 20 wins does trigger the drawdown threshold while WWLWWLWWL... etc. does NOT, even though both sequences have exactly 20 wins and 10 losses.

The big drawback with a brute force method like the one I described above is that you run quickly into a "combinatorial explosion" when the number of trials gets high enough. In the OP's original example, the number of possible permutations is 2 to the 30th power. Now anyone who knows about compounding knows that 10 doublings is a bit over a thousand (1024) so we're talking about over a billion permutations here.

2^30 = 1,073,741,824

Right. So this is why I'll be using an example with much smaller numbers. Let's see what the probability of a 3-unit drawdown would be for any period of 5 trials. Also, I'll be using the FIRST interpretation of the drawdown rule that I discussed a couple of posts ago, which is that a 3 unit drawdown only occurs if the total equity ever falls to -3 (not 3 units below some high water mark).

5 trials gives us 2^5 = 32 possible sequences. They are:

0 LLLLL
1 LLLLW
2 LLLWL
3 LLLWW
4 LLWLL
5 LLWLW
6 LLWWL
7 LLWWW
8 LWLLL
9 LWLLW
10 LWLWL
11 LWLWW
12 LWWLL
13 LWWLW
14 LWWWL
15 LWWWW
16 WLLLL
17 WLLLW
18 WLLWL
19 WLLWW
20 WLWLL
21 WLWLW
22 WLWWL
23 WLWWW
24 WWLLL
25 WWLLW
26 WWLWL
27 WWLWW
28 WWWLL
29 WWWLW
30 WWWWL
31 WWWWW

Whew! Ok, now at the beginning of each sequence I've put an index number that counts from 0 to 31. If you substitute ones for wins and zeros for losses in the sequences, you'll come out with the binary string representing the index number. This is how I keep from becoming lost whenever I have to write a list of possible sequences like that.

Now let's analyze the first sequence, LLLLL. We want to know the probability of this sequence occurring, and whether or not it trips the drawdown threshold. The probability of this sequence is simply:

(0.4)(0.4)(0.4)(0.4)(0.4) = 0.01024

We can also easily see that this sequence would trip the drawdown threshold on the third trade. So this permutation's occurrence probability gets added into our final sum.

We just do this for all 32 permutations to get the final answer. In the next post I'll show an Excel display that does this for all of them and arrives at the final answer.

ah I see, I interpreted the questions as ending up with -10 or less after 30 trials (>20 losses) not having a draw down of -10 within the 30 trials, that's tricky.

Ok, here's the full analysis for the 3 units in 5 trials example that I used. The probability of such a drawdown in 5 trials turns out to be a little over 11%.

The OP's example couldn't be done this way though because there are WAY too many permutations. That would have to be done via a program like the one I described.

Thanks for those replied, especially those who have made an effort to solve the problem. Up to now, I only certain that capitalist88 fully understand the problem. AceOfTrades and ForexQuant may have too. I don't think ForexQuant has been successful yet (perhaps I am wrong). Intu may still be a bit unclear even through his BASIC program approach has simulated correctly.
My example use 30 trials, in reality 30 is an example, I may need more than this.
Cheers thanks for all anyway.

Capitalist88 does some great work, a first-rate mind.
Your Archive of Research Notes on your site is phenomenal as well.


A healthy, thoughtful thread... thanks everyone, I've enjoyed participating.

(edited)
In response to the following findings by kk007 and ForexQuant, I do believe the "seed" for the random number generator in my program may have generated some not so random results. The program would also inexplicably "crash" often with the lower win% settings (40%, 30%, etc) when trying to test series of up to 10,000 groups of 30 trades. I had to test those in groups of 500, still... the random number generator is most likely the culprit.

Perfect! I got the same answer as you! My whole day of work!

70% ........... 0.02%
60% ........... 0.59%
50% ........... 7.08%
40% ........... 34.09%
30% ........... 76.52%
20% ........... 97.91%

I verified the answer two ways:
1. Exact value in less no. of trades
2. Simulation similar to Intu's algorithm.

My logic is similar to your one-touch logic.

Yes!!! Finally we managed to find a solution. Btw I checked the probability of touching -3 units in 5 trials by using pascal triangle model and i got an exact 11.008% as hinted by capitalist88's approach.

Anyway I just wonder why the result from intu's QBASIC code went too far compare to our calculation. I have checked his algorithm but i found no mistake.

So far my only explanation is the QBASIC random number generator is not 'truely' random. Anyone else got any other ideas?

This is the Excel VBA code, in case you are interested.

Private Sub CommandButton1_Click()
Count2 = 0
For a = 1 To 10000
Over = 0
Count1 = 0
For b = 1 To 30
If (Rnd() >= 0.6) Then
Count1 = Count1 - 1
Else
Count1 = Count1 + 1
End If
If Count1 <= -10 Then
Over = 1
End If
Next b
If Over = 1 Then
Count2 = Count2 + 1
End If
Next a
MsgBox (CStr(Count2))
End Sub

Thanks for your VBA code.

Here is my excelsheet for pascal triangle model. Have fun!