9.3 Introduction to Probability

The last statistics unit on the Level I map is probability: how we talk about uncertainty after we have means, SDs, and z-scores. It still lives in Advanced Techniques (26%). Independent OpenExamPrep teaching for this CMT Level I topic is the vocabulary of chance, the normal benchmark, and the reason technicians should not pretend every return was drawn from that benchmark.

What probability is

Probability is a number between 0 and 1 that quantifies how likely an event is. An event that cannot occur has probability 0. An event that must occur has probability 1. If P(A) is the probability of A, the probability that A does not occur is 1 − P(A) (the complement).

Two equivalent teaching stories:

• Long-run frequency. If a process can be repeated under the same conditions, P(A) is the limiting proportion of trials in which A happens. A fair coin's P(heads) = 0.5 because about half of a very long sequence of flips would be heads.
• Degree of belief (useful when you cannot flip the market 10,000 times this afternoon). You still encode uncertainty as a number in [0, 1], but you must say what information that number conditions on.

For technicians, "the probability that this breakout holds" is only a probability given a defined event and a defined sample. "Holds" must mean something countable: closes above the level for 10 sessions, or never undercuts the breakout bar's low, and so on. Vague English is not an event.

The law of large numbers

The law of large numbers (LLN) says that as the number of independent trials n grows, the sample average of a random variable converges to its expected value. A setup with a true 55% win rate can easily print 4 wins in 10 trades. In 1,000 independent trades, the observed win rate should sit near 55% if the process did not change.

LLN is why sample size matters in any backtest or in any claim that "this pattern works." It is also why a quiet 20-day SD is a noisy estimate of the volatility.

What LLN does not say: that a string of losses makes a win due. That is the gambler's fallacy. Independent trials have no memory. If trades were independent and the win probability stayed 55%, the next trade is still 55% after five losses. Markets often violate independence (below), which is a different problem: then the next trade is not the same experiment, and you should not use the 55% number blindly either.

Random variables and i.i.d.

A random variable is a numerical outcome of a random process. Daily percent return is a random variable. So is today's true range, the number of advancing issues, or a 0/1 indicator that a close was above VWAP. Discrete random variables take a countable set of values (number of inside days this week). Continuous random variables take values on an interval (a return of −0.413%).

A sequence is i.i.d. when the observations are independent and identically distributed:

• Independent: knowing yesterday's return does not change the probability distribution of today's return.
• Identically distributed: every observation is drawn from the same distribution (same mean, same SD, same shape).

Those two assumptions unlock textbook formulas: a sample mean is unbiased for μ, SD estimates σ in a standard way, and z-scores map to normal-table probabilities.

Market returns are not reliably i.i.d. Volatility clusters: a −3% day is often followed by a wide day, not by a typical quiet day (you will meet this again as historical versus implied volatility). Distributions also change with regime: the SD of a calm year is not the SD of a panic month. Consecutive overlapping returns (five-day returns sampled every day) are dependent by construction. When a stem says "assume i.i.d.," use the textbook tool. When a stem asks whether technicians should assume i.i.d. for every return series, the honest answer is no.

The normal probability distribution

The normal (Gaussian) distribution is the familiar bell curve. It is symmetric about its mean. Mean = median = mode. It is fully specified by two parameters: mean μ and standard deviation σ. Changing μ slides the bell. Changing σ widens or narrows it.

Why it shows up everywhere in exam land:

• Many sums and averages of independent pieces are approximately normal (central limit theorem), even if the pieces are not.
• Once you have a z-score, the normal table converts z into a probability if the normal model is acceptable.

A standard normal is the special case μ = 0, σ = 1. Any normal X converts to standard normal by z = (X − μ) / σ—the same z-score as in 9.1, now given a probability reading.

The normal curve's tails fall off quickly. That is a feature of the math and a bug as a model of crash-prone markets.

The Empirical Rule (68-95-99.7)

For a normal distribution, the Empirical Rule (three-sigma rule) is the memorized area:

• About 68% of observations lie within 1 SD of the mean.
• About 95% lie within 2 SD.
• About 99.7% lie within 3 SD.

Worked technician numbers. Suppose daily returns are modeled as normal with mean 0.04% and SD 1.00%.

• ~68% of days between −0.96% and +1.04%
• ~95% of days between −1.96% and +2.04%
• ~99.7% of days between −2.96% and +3.04%

A −2% day is about a −2 SD event: unusual under the model (the two tails together beyond 2 SD are about 5% of days, so each tail is about 2.5%). A −5% day is a −5 SD event. Under a normal curve, 5-sigma days are essentially once in geological time. Equity indexes have printed multiple such days in a living technician's career. That mismatch is the fat-tail story below.

Exam trap: applying 68-95-99.7 to a sample you already know is skewed or two-humped. The Empirical Rule is a normal rule. For other shapes, those percentages are wrong.

Skew

Skew (skewness) measures asymmetry.

• Zero skew: left and right tails match; the normal curve is the picture.
• Positive (right) skew: a long tail to the right. Mean is pulled above the median. A few very large positive outcomes (lottery-like payoffs, some growth names, long-option payoffs).
• Negative (left) skew: a long tail to the left. Mean is pulled below the median. Equity index returns are the textbook case: many modest up days, occasional crashes. That is why "the average day is up" and "the painful days are larger than the pleasant days" can both be true.

For a technician, skew tells you which side the surprise lives on. A negatively skewed return distribution means your left-tail stop is the one that will feel impossible when it hits. A positively skewed trading overlay (small losses, rare large wins) will look like a losing smear most weeks and still have a positive mean.

Kurtosis and fat tails

Kurtosis describes tail weight (and, in older language, how peaked the center is compared with a normal). The normal distribution is the reference: its kurtosis is 3, so excess kurtosis is kurtosis minus 3. Excess kurtosis of 0 matches the normal.

• Leptokurtic (excess kurtosis > 0): fat tails and usually a sharper peak. More tiny days and more extreme days than a normal with the same SD. This is the usual description of daily financial returns.
• Platykurtic (excess kurtosis < 0): thinner tails than normal; fewer extremes.

Fat tails mean P(|return| is huge) is larger than the Gaussian formula says. The Empirical Rule's 99.7% inside 3 SD becomes an overstatement of safety. More than 0.3% of days live beyond 3 SD. Sometimes far more.

Why technicians should not treat every return as Gaussian

Put the pieces together.

If you treat every return as i.i.d. normal, then:

• A 2-sigma stop is a ~5% event (both tails) or a ~2.5% event (one tail).
• A 3-sigma breakout is a once-in-many-months curiosity on a daily chart.
• A 5-sigma crash cannot happen, so you do not pre-plan it.

Real return samples are typically leptokurtic, often negatively skewed, and not i.i.d. Volatility clusters. Regimes shift. The chart tools you already studied—gaps, climax volume, range expansion, breakdowns—are how those fat-tail days look. They are not rounding errors in a bell curve. They are why classical pattern and volatility chapters exist.

Practical Level I consequences:

• Do not convert a z-score into a precise normal probability and then size a book as if that probability were a physical law.
• Do use mean and SD as descriptive rulers for a stated window.
• Do remember that "3-sigma" in markets is a headline, not a guarantee of rarity.
• When an item asks what the Empirical Rule would say, give 68-95-99.7. When it asks whether technicians should believe those percentages for every return series, say no—fat tails.

The next Advanced Techniques chapters (volatility, then quantitative methods) will give you other rulers: true range, implied volatility, and test design that respects sample bias. Probability here is the warning label on the Gaussian shortcut.

Key Takeaways

• Probability is in [0, 1]; LLN is about large n, not due wins
• i.i.d. is the textbook assumption; market returns often violate it
• Normal + Empirical Rule: 68% / 95% / 99.7% within 1 / 2 / 3 SD
• Skew = asymmetry; kurtosis = tail weight versus the normal
• Fat tails: do not treat every return as Gaussian when you plan for extremes