Olga Malyuk
Selected work (2)

Conditional Probability

A visual guide to understanding how new information changes probability.

Category
Visual explanation
Format
PDF, 16 pages (A4)
Language
English
Author
Olga Malyuk
Page 5, “Our sample space shrinks”: the Orange Team row of the table is crossed out in red and the Blue Team row (4 love math, 6 hate math, 10 in total) is outlined in blue, while a confused Blue Team student asks “Wait… where did everyone go?!”Page 6: the question “What percentage of the Blue Team hates math?”, the Blue Team row with the 6 circled, the calculation 6/10 = 0.6 = 60%, and a strip of 22 squares narrowed to the 10 Blue Team squares, comparing 6/22 ≈ 27% with 6/10 = 60%.
Pages 5 and 6. The Orange Team is struck out of the table, the Blue Team row becomes the whole sample space, and a strip of 22 squares narrows to 10: 6/22 ≈ 27% becomes 6/10 = 60%.

§1The concept

The guide builds conditional probability from one classroom story: 22 students, split at random into an Orange Team and a Blue Team, each asked whether they love or hate math.

Written as an introduction, it starts from the simplest case, where every student is equally likely to be chosen. It moves from P(B ∩ H) = 6/22 to P(H | B) = 6/10, introduces the notation, derives the general definition P(A | B) = P(A ∩ B) / P(B), and ends with six exercises and an answer key.

§2The learning problem

The same six students give two different answers. Picked from the whole class, a Blue Team student who hates math has probability 6/22 ≈ 27%. Picked from the Blue Team alone, the probability of hating math is 6/10 = 60%.

The story voices the tempting answer, “6/22…?”, then shows what actually changed: the sample space, which shrinks from 22 students to 10. The exercises return to two related traps: swapping the events, P(H | B) versus P(B | H), and assuming that new information can only make a probability smaller.

§3The visual approach

Colour tracks the events throughout: blue for the Blue Team and the condition B, red for students who hate math. Handwritten notes and speech bubbles sit beside the calculations.

  1. Page 7, “This is what we call a conditional probability”: all 22 students drawn as smiling or frowning faces by team, an arrow labelled with the condition “the student belongs to the Blue Team”, and below it only the Blue Team row of 4 smiling and 6 frowning faces.
    Page 7Every student is drawn as a face. Applying the condition keeps only the Blue Team row, so 22 possible students become 10, and a note explains that the Orange Team is outside the new sample space, not gone.
  2. Page 11, “Now we impose the condition B”: the class diagram with the Orange Team faded and the six Blue Team students who hate math circled, followed by the boxed formula P(H | B) = P(H ∩ B) / P(B) with arrows labelling the numerator and denominator.
    Page 11When the formula is derived, the same picture returns: everything outside B fades, and arrows tie the numerator to “the part of B that also belongs to H” and the denominator to “all of B”.
  3. Page 14, “Here’s what that looks like”: three Venn diagrams of events A and B showing all outcomes, the outcomes outside B fading, and the overlap of A and B hatched, then P(A | B) drawn as the hatched overlap over the whole circle B.
    Page 14The general case in three Venn diagrams, ending with P(A | B) drawn as a fraction made of the shapes themselves.

§4The complete guide

Page 1 of the guide: the title “Conditional Probability.” with the handwritten line “Apparently, knowing things changes things.”, a table of 22 students by team and by whether they love or hate math, and stick-figure characters from the Orange and Blue Teams.
Page 1
Pages
16
Language
English
Format
PDF, A4, 2.9 MB
Parts
The setup, Let’s explain, The notation, Let’s derive it, The general formula, Your turn