Probability & Conditional Probability

Represent uncertainty with events and probabilities, then update a probability when information about an event is known.

By the end of this lesson

You will be able to

Define an event and a sample space.
Calculate a conditional probability from observed counts.
Distinguish independence from mutually exclusive events.
Explain Bayes’ rule as probability updated by evidence.

Probability is a measure of uncertainty

An event is a set of outcomes from a sample space. For equally likely outcomes, P(A) = favourable outcomes / total outcomes. With observed data, a simple empirical estimate is count(A) / n. Probability values range from 0 to 1. Events are independent when learning that one occurred does not change the probability of the other.

rides = [0, 1, 1, 0, 1, 1, 0, 1]
observed_rate = sum(rides) / len(rides)
print(observed_rate)  # 0.625

Condition on the information you have

Conditional probability is P(A | B) = P(A and B) / P(B), provided P(B) > 0. The denominator is the condition: the population or cases where B is known to be true. Bayes’ rule reverses the condition: P(A | B) = P(B | A)P(A) / P(B). It combines a prior probability with how likely the evidence is under that event.

# A small table of observed counts
# rain: 30 rainy hours, 70 dry hours
# high_demand among rainy: 18
# P(high_demand | rain) = 18 / 30
print(18 / 30)  # 0.6

Apply what you learned

Exercise

Calculate conditional rates from a table

Use a made-up two-way table to make sure the condition is always reflected in the denominator.

Create a table with 40 rainy hours and 60 dry hours.
Record 24 high-demand rainy hours and 18 high-demand dry hours.
Calculate P(high demand | rain).
Calculate P(rain | high demand) and explain why it is different.
State whether the counts alone prove that rain causes high demand.

Done when

You calculate P(high demand | rain) = 24/40 = 0.6 and P(rain | high demand) = 24/42; your explanation says association alone does not establish causation.

Free references

Study resources

3 links

01
Conditional Probability, Clearly Explained Video
Watch this one video and identify the event used as the condition in each example.

Done
02
Conditional probability using two-way tables Article
Read the worked two-way-table example and calculate the conditional probability in both directions.

Done
03
Bayes’ Theorem, Clearly Explained Video
Watch the standalone derivation after conditional probability; focus on prior, likelihood and posterior.

Done

Check your understanding

Self-test

2 questions

Question 1

In P(A | B), what does the denominator represent?
Question 2

If A and B are independent, what is P(A | B)?

Completion check

Move on when

You can calculate a conditional probability in the correct direction and explain why conditioning changes the denominator.

Mark passed and continue Back to module