Simple Probability

When an Apothecary draws a herb at random from a jar, how likely is any particular herb? Probability measures how often something happens out of everything that could happen.

Every probability sits between 0 (impossible) and 1 (certain). Answers are normally written as fractions.

Worked Example — A Jar of Seeds

The Apothecary's jar holds 12 seed-cards. 4 are basil, 3 are thyme, 3 are rosemary, 2 are sage.

What is P(basil)?

Favourable = 4 (the basil cards)

Total = 12

P(basil) = 4/12 = 1/3  (simplify)

Another — P(not basil)?

Favourable = 12 − 4 = 8 (all the others)

P(not basil) = 8/12 = 2/3

Shortcut: P(not A) = 1 − P(A). Here 1 − 1/3 = 2/3.

Counting in words — "vowels in MONEY"

The word MONEY has 5 letters. The vowels are O and E → 2 vowels.

P(vowel) = 2/5

Venn Diagrams

An Apothecary's ledger records which patients have coughs and which have fevers. Some have both, some have neither. A Venn diagram shows all four groups at once.

Each number inside a region is a count (how many people), not a probability. To find a probability, divide by the total.

Reading the diagram

Suppose 8 patients have cough only, 5 have fever only, 4 have both, and 3 have neither.

• Total = 8 + 5 + 4 + 3 = 20
• Cough (anyone with cough) = 8 + 4 = 12
• Cough only (cough but NOT fever) = 8
• Neither = 3

Probabilities

P(cough but not fever) = 8/20 = 2/5

P(neither) = 3/20

Conditional — P(fever given cough)

"Given" narrows the world. Only look at the people with cough (12 of them). Of those, 4 also have fever.

P(fever | cough) = 4/12 = 1/3

Two-way Tables

A two-way table cross-classifies data by two categories. Rows for one, columns for the other. Every cell counts how many fall into both categories.

Worked Example

Forty Apothecary apprentices are surveyed. 22 are men, 18 are women. 15 passed their herbalism exam, and 9 of those were women.

Use row and column totals to fill the missing cells. Grand total sits in the bottom-right corner.

Reading probabilities

P(passed) = 15/40 = 3/8

P(woman) = 18/40 = 9/20

P(woman AND passed) = 9/40

Conditional — P(passed given woman)

"Given woman" means we look at the women's row only: 18 women, of whom 9 passed.

P(passed | woman) = 9/18 = 1/2

Checking independence

If P(passed | woman) = P(passed), the events are independent (gender doesn't affect passing). Compare:

P(passed) = 15/40 = 0.375

P(passed | woman) = 9/18 = 0.5

These aren't equal, so gender DOES affect passing — being a woman raised the probability from 37.5% to 50%.

Tree Diagrams

When an experiment has multiple stages — draw one card, then another; flip two coins; pick two marbles — a tree diagram maps every possible outcome.

Worked Example

A pouch holds 6 stones: 4 obsidian, 2 moonstone. Draw one, put it back, then draw a second.

Along the branches — multiply

To find the probability of a whole path through the tree, multiply along the branches:

P(obsidian, obsidian) = 4/6 × 4/6 = 16/36 = 4/9

P(obsidian, moonstone) = 4/6 × 2/6 = 8/36 = 2/9

Between branches — add

For "same colour both times":

P(OO) + P(MM) = 16/36 + 4/36 = 20/36 = 5/9

"At least one" — use the complement

"At least one obsidian" is hard to count directly. Easier: 1 minus the probability of NONE.

P(at least one O) = 1 − P(MM) = 1 − 4/36 = 32/36 = 8/9

Building a Venn Diagram

Chapter II showed you how to READ a Venn diagram. Now you'll BUILD one from words.

Worked Example

From a class of 25 apprentices: 14 can brew potions, 11 can craft amulets, and 6 can do both.

Verify with probabilities

Once built, reading probabilities is the same as Chapter II:

P(potions only) = 8/25

P(both) = 6/25

P(neither) = 6/25

Listing the Sample Space

For complex scenarios with several choices in sequence, the cleanest tool is to list EVERY possible outcome systematically. This is the sample space.

Worked Example — a Traveller's Choices

An apothecary travels to gather rare herbs. Choices:

• Destination: mountain (M) or forest (F)
• Transport: horse (H) or cart (C)
• Bag: satchel (S) or basket (B)

Restriction: you can't take a basket on a horse (too fragile).

List every allowed combination:

Total: 6 allowed combinations (forbidden: basket with horse, i.e., M-H-B and F-H-B).

Reading probabilities

If every allowed combination is equally likely:

P(forest) = 3/6 = 1/2

P(basket) = 2/6 = 1/3

Conditional — P(forest given basket)

"Given basket" → restrict to just the basket outcomes: M-C-B and F-C-B. Of those, 1 goes to forest.

P(forest | basket) = 1/2

The Apothecary's Journal