Probability basics · RiskAtoms Academy

By the end you will be able to

1Describe outcomes and events, and find a probability between 0 and 1 by counting equally likely outcomes.
2Use the complement: P(not A) = 1 − P(A).
3Combine events with 'and' (multiply, for independent events) and 'or' (add, for mutually exclusive events, minus any overlap).
4Tell probability as a long-run frequency apart from probability as a judgement.

Why you need it

Will a borrower repay? Will the system go down this year? Nobody can say for sure, but you can say how likely it is. Probability is the number that does that.

These Core Path lessons use it:

L2.1.02: Risk and uncertainty
Telling risk (where you can put a probability on the outcomes) apart from uncertainty (where you cannot), and frequency apart from judgement.
L2.4.03: Assessing risks: likelihood and impact
Likelihood on a risk heat map is a probability, often given as '1 in 20 years' or 5% a year, and combined with and/or rules.

It is also the starting point for every other atom in this block, beginning with M-C02: Expectation.

The idea

A number from 0 to 1

Roll a fair die. Six things can happen: the faces 1 to 6. Each is an outcome. Usually you care about a group of outcomes, such as “an even number” (2, 4 or 6). That group is an event.

A probability puts a number on how likely an event is. It always sits on a scale from 0 to 1. Zero means it cannot happen; 1 means it is certain. In between, bigger means more likely.

The probability scale, with events from one fair die

What does “one sixth” really mean? You cannot roll one sixth of a six. It means that if you rolled the die again and again, about one roll in six would be a six. The more you roll, the closer the share gets. This is probability as a long-run frequency, and you can watch it happen in the Try it below.

The rules

First, the notation. We call an event by a capital letter, such as $A$ , and write its probability as $P(A)$ .

The complement. Sometimes it is easier to work out the chance that something does not happen. The complement rule turns one into the other.

“And”: both happen. When two events are independent, multiply.

“Or”: at least one happens. When two events are mutually exclusive, add. When they can overlap, take the overlap off once.

The rules, for reference

0 \le P(A) \le 1 \qquad P(\text{not } A) = 1 - P(A)

P(A \text{ and } B) = P(A)\times P(B) \quad \text{(independent)}

P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)

The ‘or’ rule always works; for mutually exclusive events the last term is 0.

Equally likely outcomes: $P(A) = \frac{\text{outcomes in } A}{\text{all outcomes}}$ . A probability of $p$ is the same as $100p\,\%$ : 0.03 is 3%.

Two ways to read a probability

Some probabilities come from counting: a fair die, or a long record of similar cases. Others are a judgement: a considered view of how likely something is when there is nothing to repeat. Both are numbers from 0 to 1, and all the rules above apply to both. But a judgement deserves more humility, and should change as evidence arrives. Decide which kind each one is, then reveal.

Frequency or judgement?

Guess, then reveal

“A fair coin lands heads with probability 0.5.”
“Of 2,000 similar loans, 60 defaulted within a year, so we use 3% a year.”
“The project team thinks there is a 20% chance the new system goes live late.”
“The river wall is built for a 1-in-100-year flood.”

Try it

Two dice, a red one and a blue one, give 36 equally likely outcomes: one square each. Pick two events and a way to combine them. Count the lit squares to get the exact probability. Then roll and watch the share of rolls in the event settle near that count. Things to try: “The total is 7” and “A double” with A and B (they never overlap); “The red die is even” and “The blue die shows 5 or 6” (independent); “At least one six” with not A.

Try itTwo dice: events and long-run frequency

Open the probability calculator →

Event A Event B

Highlight

Each square is one outcome; the number is the total. Dot: in A. Ring: in B.

Exact probability, by counting squares

6/36 = 1/6 ≈ 0.1667

A and B can happen together?No: mutually exclusive

Independent?No

Rolls0

Last roll (red, blue)–

Rolls in the event0

Share of rolls in the event–

Exact probability0.1667

The share of rolls that land in the event (line) against the exact probability (dashed). Dice set 1.

The rolls come from a seeded random number generator: the same dice set always gives the same rolls, so you can repeat an experiment. “Start again” moves to a new set.

Worked examples

1. Two borrowers, both or neither

Step 1 of 4

1
The set-up
A lender in Singapore has two unrelated borrowers. Each has a 2% chance of defaulting this year, and we assume they are independent. What is the chance both default? Neither? At least one?

2. Outcomes that cannot happen together

Step 1 of 4

1
The set-up
A company’s credit rating next year will do exactly one of four things. Each outcome has a probability (illustrative figures):
Next year Probability
Upgraded 0.08
Unchanged 0.85
Downgraded 0.06
Defaults 0.01
Total 1

Next year	Probability
Upgraded	0.08
Unchanged	0.85
Downgraded	0.06
Defaults	0.01
Total	1

3. At least one outage: why adding goes wrong

Step 1 of 4

1
The set-up
A firm relies on 5 separate IT suppliers. Each has a 3% chance of a serious outage in a year, independently of the others. What is the chance that at least one has an outage?

Common mistakes

Each line below is a tempting slip. Spot what is wrong, then check.

Reaction

Check what stuck

Question 1 · objective 1

You roll two fair dice. What is the probability that the total is 7? Give a decimal to three places.

Question 2 · objective 2

A borrower has a 3% chance of defaulting this year. What is the probability that it does not default?

Question 3 · objective 3

Two independent events have probabilities 0.1 and 0.2. What is the probability that both happen?

Question 4 · objective 3

You roll one die. Which pair of events is mutually exclusive?

AEven, and greater than 3BA 1, and a 6CEven, and a 2DOdd, and less than 4

Question 5 · objective 4

An analyst says: 'I put the chance that the new office opens on time at 70%.' What kind of probability is this?

AA long-run frequency counted from many identical repeatsBA judgement: a degree of belief based on what the analyst knowsCNot a probability, because it cannot be checkedDA certainty, because it is above 50%

Glossary

Outcome: One thing that can happen, such as the die showing 4. The full list of outcomes covers everything possible.
Event: A set of outcomes you are interested in, such as 'an even number'. It happens if any one of its outcomes happens.
Probability: A number from 0 (impossible) to 1 (certain) saying how likely an event is. Often written as a percentage: 0.05 is 5%.
Complement: 'Not A': every outcome that is not in A. P(not A) = 1 − P(A).
Independent events: Two events are independent when knowing that one happened does not change the chance of the other. Then P(A and B) = P(A) × P(B).
Mutually exclusive: Two events are mutually exclusive when they cannot both happen. Then P(A or B) = P(A) + P(B).
Long-run frequency: Probability read as the share of times an event happens over many repeats of the same situation.
Judgement (subjective probability): Probability read as a degree of belief, given what you know, when there are no repeats to count. Also called subjective probability.