Probability, Sets, Venn Diagrams and Two-Way Tables

In short: Probability is how likely an event is: favourable outcomes divided by total outcomes, from 0 (impossible) to 1 (certain). Venn diagrams and two-way tables organise events, and $P (A \cup B) = P (A) + P (B) - P (A \cap B)$ .

Starting point: Years 9 and 10

Basic probability

For equally likely outcomes:

$P (event) = \frac{number of favourable outcomes}{total number of outcomes}$

Probabilities go from 0 (impossible) to 1 (certain). The list of all possible outcomes is the sample space.

The complement of an event is "it does not happen":

$P (not A) = 1 - P (A)$

Venn diagrams

Fill the middle first: 5 do both. Then 18 − 5 = 13 sport only, 12 − 5 = 7 music only, and 30 − 25 = 5 neither.

So $P (sport but not music) = \frac{13}{30}$ and $P (neither) = \frac{5}{30} = \frac{1}{6}$ .

Two-way tables

The same data can go in a table with totals.

	Music	No music	Total
Sport	5	13	18
No sport	7	5	12
Total	12	18	30

Every row and column must add up to its total, which is a good way to check your work.

Probability and sets

Basics

Favourable over total
0 to 1
Complement: 1 - P(A)

Venn diagrams

Overlap is "and"
Inside either is "or"

Two-way tables

Rows and columns add up

Set notation

∪ union, ∩ intersection
A' complement

Rules

Addition rule
Mutually exclusive
Conditional P(A | B)
Independent events

Year 11Conditional probability and independence

$P (A ∣ B)$ means "the probability of $A$ , given that $B$ has happened." You restrict attention to $B$ only.

$P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}$

Example. Given a student plays music, what is the chance they play sport? Only look at the 12 music students. 5 of them play sport, so $P (sport ∣ music) = \frac{5}{12}$ .

Events are independent if one does not change the chance of the other:

$P (A \cap B) = P (A) \times P (B)$

In the survey, $P (sport) \times P (music) = 0.6 \times 0.4 = 0.24$ , but $P (both) = \frac{5}{30} \approx 0.17$ . They are not independent.

Testing independence from a union. If $P (A) = 0.5$ , $P (B) = 0.4$ and $P (A \cup B) = 0.7$ , the addition rule gives $P (A \cap B) = 0.5 + 0.4 - 0.7 = 0.2$ . Since $0.5 \times 0.4 = 0.2$ too, $A$ and $B$ are independent.

Flipping a condition. To get $P (A ∣ B)$ from $P (B ∣ A)$ , go through the overlap: $P (A \cap B) = P (B ∣ A) P (A)$ , then divide by $P (B)$ . With $P (B ∣ A) = 0.5$ , $P (A) = 0.3$ and $P (B) = 0.25$ : $P (A \cap B) = 0.15$ , so $P (A ∣ B) = 0.6$ .

Complements of independent events are independent too: $P (A^{'} \cap B) = P (A^{'}) P (B)$ . And from a Venn diagram, $P (A^{'} \cap B) = P (A \cup B) - P (A)$ , the part of the union outside $A$ .

Finding an unknown. A bag has $n$ discs, 5 of them red. If two are drawn without replacement and $P (both red) = \frac{2}{9}$ , then $\frac{5}{n} \times \frac{4}{n - 1} = \frac{2}{9}$ , so $n^{2} - n - 90 = 0$ and $n = 10$ .

How many tries? If each test finds a carrier with probability 0.2, the chance of at least one in $n$ tests is $1 - 0. 8^{n}$ . For more than 90%: $0. 8^{n} < 0.1$ , so $n > \frac{ln 0.1}{ln 0.8} \approx 10.3$ , which means 11 tests.

Quick summary

$P (event) = \frac{favourable}{total}$ and $P (not A) = 1 - P (A)$ .
In a Venn diagram, fill the overlap first. "And" is the overlap. "Or" is anything in either circle.
Two-way tables show the same counts with row and column totals.
$P (A \cup B) = P (A) + P (B) - P (A \cap B)$ .
$P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}$ . Independent if $P (A \cap B) = P (A) P (B)$ .

Quiz

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A fair die is rolled. What is the probability of rolling a number greater than 4?
1. A $\frac{1}{6}$
2. B $\frac{1}{3}$
3. C $\frac{1}{2}$
4. D $\frac{2}{3}$
Show answer
B: $\frac{1}{3}$
5 and 6: two out of six.
The probability of rain tomorrow is 0.35. The probability of no rain is:
1. A 0.35
2. B 0.65
3. C 0.5
4. D 1.35
Show answer
B: 0.65
$1 - 0.35$ .
In a class of 25, 15 like pizza, 12 like pasta, and 6 like both. How many like pizza only?
1. A 15
2. B 9
3. C 6
4. D 3
Show answer
B: 9
$15 - 6 = 9$ .

Probability, Sets, Venn Diagrams and Two-Way Tables

Starting point: Years 9 and 10

Basic probability

Venn diagrams

Two-way tables

Quick summary

Quiz

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