Part 1 · Probability & Statistics

1.1 What is probability?

Dice frequencies and “degrees of belief”

9-minute readPrerequisite: 0.2 Reading math notation (if formulas make you nervous)▶ Video coming soon

What you will learn

Explain probability as “how likely something is, expressed as a number from 0 to 1”
Explain the three ways of thinking about probability — counting, frequency and degree of belief — and when each one fits
Calculate simple probabilities with the complement, addition and multiplication rules, and see why the assumption of “independence” can be risky

Build intuition

A “probability” is a number from 0 to 1 (0% to 100%) that expresses how likely an event is. 0 means “never happens” and 1 means “always happens”; most events fall somewhere in between.

There are, however, three broad ways of deciding how likely something is.

Way of thinking	Example	When it fits
Counting	The probability of rolling a 6 is 1 out of 6 possible outcomes: 1/6	When the mechanism is symmetric and every outcome is equally likely
Frequency	A detector detects the target in 900 of 1,000 tests → a 90% detection rate	When a test can be repeated many times, or plenty of data exist
Degree of belief	“It is unlikely that this software update introduced a new bug”	For one-off events, or when data are still scarce (updated as evidence comes in)

Counting and frequency are connected. If you keep rolling a die and track “the share of sixes”, it swings widely at first but approaches 1/6 as the number of rolls grows.

The share of sixes as a die is rolled again and again (a simulation). In this example not a single six came up in the first 10 rolls (0%), and the share was still 9% after 100 rolls, but it reached 16.7% after 1,000 — close to the true probability of 1/6.

In safety work, the difference between “frequency” and “degree of belief” is drawn as two kinds of uncertainty. Uncertainty that is built into the mechanism itself and does not go away with more data — the scatter of dice or of measurements — is called aleatoric uncertainty; uncertainty that comes from not yet knowing, or not having enough data, is called epistemic uncertainty. The first is measured by frequency; the second is handled as a degree of belief and reduced as evidence accumulates. The uncertainty in an AI’s judgements (3.7) and the “unknown hazards” of SOTIF (5.2) can be sorted out with this distinction.

This tendency to approach the true probability with many repetitions is the law of large numbers, covered in detail in 1.5. Conversely, pinning down a probability from just a few results can be badly wrong.

Check with math

① The basic rules of probability

The probability of an event A is written $P(A)$. Every probability lies between 0 and 1, and the probabilities of all possible outcomes add up to 1. So the probability that A does not happen (the complement, $\bar{A}$) is found simply by subtracting from 1.

$$0 \le P(A) \le 1, \qquad P(\bar{A}) = 1 - P(A)$$

② Addition: “A or B”

When two events cannot happen at the same time (mutually exclusive), the probability that either happens is found by addition. The probability of rolling a 1 or a 2 is 1/6 + 1/6 = 1/3. When they can happen together, subtract the overlap once.

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

③ Multiplication: “A and B”, and independence

When one event happening does not change how likely the other is (independent), the probability that both happen is found by multiplication.

$$P(A \cap B) = P(A)\,P(B)$$

Example: combine a camera that misses a pedestrian 1% of the time with a LiDAR that also misses 1% of the time, so that detection by either one is enough. If the two are independent, the probability that both miss drops to 0.01 × 0.01 = 0.0001 (once in 10,000). This is the idea behind redundancy. But this multiplication only holds when they really are independent (see “Connection to automated driving”).

④ The probability of “at least once”

Even when the probability p per trial is small, the probability that something happens “at least once” grows with the number of trials n. Find the probability that it “never happens” by multiplication, then subtract from 1.

$$1 - (1 - p)^n$$

A dangerous situation that arises in 0.01% of drives (once in 10,000) occurs at least once over 10,000 drives with probability $1 - (1 - 0.0001)^{10000} \approx 0.63$ — about 63%. This is the flip side of $e^{-1} \approx 0.37$ (the probability that it never happens) from 1.8.

Advanced: putting a number on common causes (the β factor)

In ③ we saw that multiplication only works when the events are independent. So how much larger does the probability of both missing become when there is a common cause? A simple model used in functional safety is the β factor. Of one sensor’s misses (probability p), a fraction β is assumed to come from causes — such as dense fog — that make the other sensor miss at the same time. The remaining (1 − β) occur independently. The probability that both sensors miss is then approximately

$$P(\text{both miss}) \approx \beta p + \{(1-\beta)\,p\}^2$$

With p = 1% and β = 10%, this is 0.1 × 0.01 + (0.9 × 0.01)² ≈ 0.0010 + 0.0001 ≈ 0.11% — about 11 times the 0.01% obtained by assuming independence. Even with β = 2% it is about 0.03%, three times as large. A small share of common causes is enough to make the multiplied result meaningless.

β gets smaller the more “differences” there are: different sensing principles (camera and radar), separate power supplies, separate mounting positions. Standards such as IEC 61508 give methods that score such design differences to estimate β (only the idea is summarized here). Searching for triggering conditions that weaken several sensors at once in SOTIF (5.3), and making sure the two legs of a safety-case argument do not rest on a shared assumption (5.10), are both about keeping this β small.

Summary

A probability expresses how likely something is, from 0 to 1. It can be decided by counting, by frequency, or as a degree of belief
The complement is 1 minus the probability; “or” means addition (for mutually exclusive events); “and” means multiplication (for independent events)
Multiplication assumes independence. Common causes make it easy to overestimate the benefit of redundancy

Quick quiz

Q1. What is the probability of rolling a multiple of 3 with one die?

1/3. The multiples of 3 are 3 and 6 — 2 outcomes out of 6 — so 2/6 = 1/3.

Q2. Two sensors, each missing a target 1% of the time, are combined so that detection by either is enough. What is the probability that both miss, and when does this calculation fail?

If they are independent, 0.01 × 0.01 = 0.0001, i.e. 0.01% (once in 10,000). But when something degrades both at once — dense fog or heavy rain, for example — they are not independent, and the real probability is larger.

Q3. A dangerous situation arises in 0.01% of drives. What is the probability that it occurs at least once in 10,000 drives?

About 63%. The probability that it never occurs is $(1 - 0.0001)^{10000} \approx 0.37$, so 1 − 0.37 ≈ 0.63.

Q4. Which of the three ways of thinking is closest to a weather forecast’s “70% chance of rain”?

Frequency. Japan’s Meteorological Agency defines it to mean that when a 70% forecast is issued 100 times, it rains 1 mm or more on about 70 of those occasions.