Correct. But if you do not specify the order of the events, but just their number, the probabilities change. In the first example, the event is {heads occur 4 times}. P{heads occurs 5 times in 5 trials} = 5/32 In the second example, the event is {heads occurs 2 times} P{heads occurs 2 times in 5 trials} = 10/32
This is a good point I believe. Something along these lines is mentioned in the paper below. IMO, this paper is a must read. It shows how r:r and winning odds are related but it also discusses how volatility contributes to uncertainty in the case of position and trend following systems where your winning odds may not be high enough to take in account random effects (pages 4 and 5). http://www.tradingpatterns.com/Literature/profitability.pdf
Probability when flipping a coin at each and any instant is 50/50. Treat the 5 coin flips in a row as a hypothetical trade sequence. Your odds of nailing any specific sequence is exactly 1 in 32. ie. h-h-h-h-h or t-t-t-t-t or t-h-h-t-h etc. Consider this Simple bet: You bet $500 that over the next 5 coin flips it will not be heads 5 times in a row. Statistically you should win this bet 31 out of 32 times (97%). If you simply bet $100 each toss over 32 sequences. T = Net $100 H-T = $0 H-H-T = - $100 H-H-H-T = - $200 H-H-H-H-T = - $300 H-H-H-H-H = - $500 Statistically: 16 x $100 8 x $0 4 x $-100 2 x $-200 1 x -$300 1 x - 500 ======= Net $0 Problem is you have very little rewards, statistically you make a little money on 50% of your bets, break even on 25% and lose on 25%. and this is with 97% odds of winning. But unlike Vegas this table has virtually no limits and you can purchase insurance (options) to cover this 1 specific bust scenario. How can you size your wagers to absorb the inevitable 5 in a row loss and maximize your payouts? Typical Averaging Down Scenario: Adjust your bets for each coin flip: Coin 1 - $ 100 Coin 2 - $ 300 Coin 3 - $ 700 Coin 4 - $1500 Coin 5 - $3100 T = $100 H-T = $200 H-H-T = $300 H-H-H-T = $400 H-H-H-H-T = $500 H-H-H-H-H = - $5700 Statistically still a net 0 bet: 16 x $100 8 x $200 4 x $300 2 x $400 1 x $500 1 x -$5700 ========= Net = 0 But if you purchased $1000 of options insurance that covers up to 32 sequences and pays out $10000 in the event of a bust: You risk reward profile changes dramatically: $2200 profits is your worse case (32 - $100 pops less $1000 insurance costs). Obviously this is hypothetical but the general concept is to trade high probability sequences, hedge the risk and size your trades to maximize profits.
No, the probability of getting two heads in a sequence of five tosses is not the same as the probability of getting any sequence in 5 tosses. In general the probability of getting k heads in n tosses is not the same as the probability of the outcome of any n tosses. This affects drawdown probability, as the probability of getting a number of losers in a row is always lower than getting the same losers spread in the same number of trades. Someone would think this is to their advantage but it turns out the equity reduction is always the same. Think about it 2 winners followed by 3 losers 1 winner then 1 loser then 1 winner then 2 losers The end result is the same although the probability is much different. This should ring some bells. There are some deep conceptual problems in applying a priori probability to the real world.
you're almost right. unfortunately a volatility projection is built into the pricing in different ways, thus even you use it to define exits you're still not achieving an edge per se. however, using IV levels (in the options) to define your stop loss and profit targets in linear instruments trades such as futures or stocks, i.e. using NDX IV to define a stop loss on a NQ trade, might increase the efficiency of your long term results. this path you want to undertake has merits, i assure you but dont expect to solve puzzle with that (just).
True, but now we're getting into the law of large numbers theorem... which tells us what we can expect (with the always useful SD helping our confidence) over the time of the trial. Although it sounds a bit odd, the theorem will tell us what to expect over time, but not at any specific time. The coin is still the same coin, and the flip of it is still independent regardless of prior results. A pretty funny assignment one of my stats professor use to give out was one where he asked all students to flip 400 coins and record the results in excel.... it was funny only because he could actually tell who did it (who flipped a coin 400 times and recorded it) vs. those who fudged it and wrote down what they thought would be "random". The streaks of heads/tails in a row in real trials was almost twice as much as when people made it up.... the coin can stay irrational longer than you can remain solvent