Probability Diagrams

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Notes

Two-Way Tables

**Two-way tables** compare two characteristics (e.g., year group and language).
Add row and column totals, including an **overall total** in the bottom-right corner.
For a random selection from the whole group, probability $= (number$ in category) / (overall total).
For a random selection from a specific category, denominator is that category's total.
Check that row and column totals sum to the overall total to avoid errors.

Probabilities from Venn Diagrams

If the Venn diagram shows **frequencies**, probability $= (sum$ of frequencies in region) / (total frequency).
If the diagram shows **individual elements**, probability $= (number$ of elements in region) / (total elements).
For conditional probability (e.g., P(B given A)), use only elements in A as the denominator.
When filling numbers, some given totals may need to be split between overlapping regions (e.g., '10 have a cat, 6 have both cat and dog' means 4 have cat only).

Probability Tree Diagrams

Tree diagrams show outcomes of repeated experiments with two or more stages.
Write probabilities on each branch; $P(not A) = 1 -$ P(A); probabilities on each pair of branches sum to 1.
Multiply along branches to find P(first outcome **and** second outcome).
Add probabilities of separate cases for 'or' events (e.g., P(AA or $BB) = P(AA) + P(BB))$ .
All final probabilities sum to 1; for 'at least one', use $1 - P(none)$ .
**Conditional probabilities** appear on later branches when outcomes depend on previous ones (e.g., without replacement).
In 'without replacement', denominators decrease by 1 on the second set of branches.

Combined Probabilities

**And** means multiply: P(A and $B) = P(A) \times P(B)$ .
**Or** means add: P(AA or $BB) = P(AA) + P(BB)$ .
Rephrase questions using 'and'/'or' to apply the rules.
**Independent events** do not affect each other (e.g., dice rolls); 'without replacement' events are not independent.
$P(not A) = 1 - P(A)$ can simplify 'at least one' calculations.

Two-Way Table Example

Venn Diagram Example

Probability Tree Diagram

Combined Probabilities Example (Without Replacement)

Practice questions

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1.A two-way table shows the number of students in Year 12 and Year 13 studying Spanish or German. There are 25 Year 12 students (15 Spanish, 10 German) and 30 Year 13 students (5 Spanish, 25 German). What is the probability that a randomly selected student from the college studies Spanish?
Medium
- A $\frac{20}{55}$
- B $\frac{15}{55}$
- C $\frac{5}{55}$
- D $\frac{35}{55}$
2.In a Venn diagram, the universal set ℰ has 11 elements. Set A contains 5 elements, set B contains 4 elements, and the intersection A ∩ B contains 2 elements. What is P(A ∩ B)?
Easy
- A $\frac{2}{11}$
- B $\frac{5}{11}$
- C $\frac{4}{11}$
- D $\frac{7}{11}$
3.A tree diagram shows the probability of rain $(\frac{1}{3})$ and no rain $(\frac{2}{3})$ . If it rains, Amira goes fishing with probability 3/5; if not, she goes with probability 3/4. What is the probability that on any day Amira goes fishing?
Medium
- A $\frac{7}{10}$
- B $\frac{1}{3}$
- C $\frac{2}{5}$
- D $\frac{3}{5}$
4.A bag contains 15 red beads and 10 yellow beads. Ariana picks a bead at random, records its colour, and replaces it. She then picks another bead at random. What is the probability that she picks two red beads?
Medium
- A $\frac{9}{25}$
- B $\frac{3}{5}$
- C $\frac{2}{5}$
- D $\frac{3}{10}$
5.ℰ $= {odd$ numbers less than 30}. $A = {3, 9, 15, 21, 27}$ . $B = {5, 15, 25}$ . A number is chosen at random from ℰ. What is the probability that the number is in A ∪ B?
Hard
- A $\frac{7}{15}$
- B $\frac{8}{15}$
- C $\frac{6}{15}$
- D $\frac{5}{15}$
6.In a Venn diagram, the numbers 1 to 14 are placed. A ∩ $B = {6, 10}$ . $B' = {1,2,3,4,5,7,8,9,11,13,14}$ . What is the probability that a number chosen at random from 1 to 14 is in A but not in B?
Medium
- A $\frac{5}{14}$
- B $\frac{3}{14}$
- C $\frac{2}{14}$
- D $\frac{4}{14}$
7.On any Saturday, the probability that Arun plays football is 3/4. The probability that Bob plays football is 2/5. Assuming independence, what is the probability that both play football on a Saturday?
Medium
- A $\frac{3}{10}$
- B $\frac{6}{20}$
- C $\frac{1}{2}$
- D $\frac{7}{20}$
8.Machine A makes bottles with a 0.02 probability of being faulty. Machine B makes bottles with a 0.05 probability of being faulty. One bottle is taken at random from each machine. What is the probability that at least one bottle is faulty?
Hard
- A0.069
- B0.001
- C0.07
- D0.0694

Summary

## Probability Diagrams – Quick Revision ### 1. Two-Way Tables - Display frequencies for two categories. - Add row and column totals; overall total in bottom-right. - Probability = (number in region) / (total). - For conditional probability (e.g., given a row), denominator is the row total. ### 2. Venn Diagrams - Each region shows frequency or elements. - **Key notation:** - $A \cup B$: union (in A or B or both). - $A \cap B$: intersection (in both). - $A'$: complement (not in A). - Probability = (count in region) / (total). - Conditional: given in A, probability of B = $\frac{P(A \cap B)}{P(A)}$. ### 3. Probability Tree Diagrams - Branches show outcomes; probabilities on each branch. - **Rules:** - P(not A) = 1 – P(A). - Multiply along branches for "and". - Add separate cases for "or". - For "at least one", use 1 – P(none). - Conditional probabilities: second set of branches may differ (e.g., without replacement). ### 4. Combined Probabilities - **And rule:** P(A and B) = P(A) × P(B) (for independent events). - **Or rule:** P(A or B) = P(A) + P(B) (for mutually exclusive). - Rephrase questions using "and"/"or". - Without replacement: probabilities change after each draw. ### Key Formulas - $P(\text{not } A) = 1 - P(A)$ - $P(A \cap B) = P(A) \times P(B)$ (independent) - $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ - For tree: multiply branches, add final outcomes.

Key terms

Two-way table: A table that displays frequencies for two categorical variables, with row and column totals.
Venn diagram: A diagram with overlapping circles inside a rectangle (universal set) showing relationships between sets.
Union (∪): The set of elements that are in A or B or both.
Intersection (∩): The set of elements that are in both A and B.
Complement ('): The set of elements not in the given set.
Tree diagram: A diagram with branches representing sequential events, with probabilities on each branch.
Conditional probability: The probability of an event given that another event has occurred; e.g., P(A|B).
Independent events: Events where the outcome of one does not affect the outcome of the other.
Combined probability: Probability of successive events, using multiplication for 'and' and addition for 'or'.

Slides

Probability Diagrams Overview

Probability diagrams include tree diagrams, Venn diagrams, and two-way tables.
They help visualise and calculate probabilities for combined events.
We will cover each type and how to use them to find probabilities.

Two-Way Tables

A two-way table compares two characteristics, e.g., year group and language studied.
Add row and column totals, including an overall total.
Probabilities are found by dividing the relevant cell or total by the overall total (or a specific row/column total if restricted).

In a two-way table of 60 students, 20 study Spanish and 15 are in Year 12. If 8 Year 12 students study Spanish, what is the probability a randomly chosen student studies Spanish and is in Year 12?

$\frac{8}{60}$
$\frac{20}{60}$
$\frac{15}{60}$
$\frac{8}{15}$

Probabilities from Venn Diagrams

Venn diagrams show sets and their intersections.
To find probability, count elements in the region of interest and divide by total elements.
For conditional probability (e.g., given A), restrict the denominator to the number in set A.

In a Venn diagram with total 11 elements, if A has 5 elements and the intersection of A and B has 2, then P(A but not $B) = ____ / 11$ .

Probability Tree Diagrams

Tree diagrams show outcomes of successive events.
Probabilities on branches from the same point add to 1.
Multiply along branches for 'and', add between branches for 'or'.

Which of these are true about tree diagrams?

Probabilities on each pair of branches add to 1.
To find P(A and B), you add the probabilities on the branches.
Tree diagrams can handle conditional probabilities.

Tree Diagram Example: Traffic Lights

First set: $P(Green)=\frac{5}{7}, P(Red)=\frac{2}{7}$ .
Second set: $P(Green)=\frac{8}{9}, P(Red)=\frac{1}{9}$ .
$P(both red) = (\frac{2}{7})\times (\frac{1}{9}) = \frac{2}{63}$ .

Using the traffic lights tree diagram, what is P(at least one red)?

$\frac{23}{63}$
$\frac{40}{63}$
$\frac{2}{63}$
$\frac{5}{63}$

Tree Diagrams: Without Replacement

When items are not replaced, probabilities on second branches change.
Denominators decrease by 1 after each draw.
Example: 7 guinea pigs, 3 rabbits. First pick: $P(G)=\frac{7}{10}$ . Second pick after $G: P(G)=\frac{6}{9}$ .

From 7 guinea pigs and 3 rabbits, without replacement, $P(two rabbits) = (\frac{3}{10}) \times (____/9)$ .

Combined Probabilities

Use 'and' rule: P(A and $B) = P(A) \times P(B)$ for independent events.
Use 'or' rule: P(A or $B) = P(A) + P(B)$ for mutually exclusive events.
For 'at least one', use $1 - P(none)$ .

A bag has 3 blue and 8 red counters. With replacement, what is P(both red)?

$\frac{64}{121}$
$\frac{24}{121}$
$\frac{8}{11}$
$\frac{16}{121}$

Venn Diagram Example: Class of 30

15 study Spanish, 3 study both Spanish and German, 7 study neither.
So 12 study Spanish only, 8 study German only.
P(Spanish but not $German) = \frac{12}{30} = \frac{2}{5}$ .

In the class of 30, P(studies German $only) = ____ / 30$ .

Two-Way Table Example: Art Class

60 children in two classes, activities: colouring, painting, clay, sketching.
Fill in given numbers, then use totals to find missing values.
$P(colouring) =$ 13/60; P(class A and $sketching) = \frac{8}{60} = \frac{2}{15}$ .

From the art class two-way table, what is P(painting | class B)?

$\frac{12}{30}$
$\frac{12}{60}$
$\frac{20}{60}$
$\frac{8}{30}$

Summary

Two-way tables: use totals to find probabilities.
Venn diagrams: count elements in regions.
Tree diagrams: multiply along branches, add between branches.
Combined probabilities: 'and' multiply, 'or' add, 'at least one' complement.

Which statements are correct?

P(A and $B) = P(A) \times P(B)$ always.
In a tree diagram, probabilities on branches from the same point sum to 1.
Without replacement, probabilities on second branches change.

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