Speak Maths · University

Probability: how likely, and given what

A chance is a number from 0 to 1. Seven signs say how likely something is, how that changes when you learn more, and what to expect on average.

Draft: written by an AI helper and not yet checked by a teacher.

The alphabet

Read each sign aloud, the way it is written under it.

$P (A)$ Say: P of A, the probability of AMeans: the chance that A happens, from 0 (never) to 1 (certain)
$P (A \cap B)$ Say: P of A and B, the probability of A intersection BMeans: the chance that both A and B happen
$P (A ∣ B)$ Say: P of A given BMeans: the chance of A once you know B has happened
$A ⊥ B ⟺ P (A \cap B) = P (A) P (B)$ Say: A and B are independent: P of A and B equals P of A times P of BMeans: knowing B tells you nothing about A
$E [X]$ Say: the expectation of X, or E of XMeans: the average value of X over very many repeats
$Var (X)$ Say: the variance of XMeans: the average squared distance of X from its mean: how spread out it is
$X \sim N (μ, σ^{2})$ Say: X is normally distributed with mean mu and variance sigma squaredMeans: X follows the bell curve centred on mu, with spread sigma

The story: Rain and umbrellas

Subtitles:

Tap a line to see its words.

$P (R) = \frac{30}{100} = 0.3$ Over 100 days, it rained on 30. The chance of rain on a day is 0.3.
$P (U) = \frac{40}{100} = 0.4$ On 40 of those days, the umbrella came along.
$P (R \cap U) = 0.27$ On 27 days it rained and the umbrella was there.
$P (U ∣ R) = \frac{0.27}{0.3} = 0.9$ Of the 30 rainy days, the umbrella was along on 27: given rain, the chance of an umbrella is 0.9.
$P (R ∣ U) = \frac{0.27}{0.4} = 0.675$ Of the 40 umbrella days, it rained on 27: given an umbrella, the chance of rain is 0.675.
$E [X] = 7 \times 0.3 = 2.1$ Over a week, let X be the number of rainy days. On average there are 2.1.
$Var (X) = 7 \times 0.3 \times 0.7 = 1.47$ If each day’s weather is independent of the others, the spread of X, its variance, is 1.47.
$T \sim N (12, 4)$ The walk to the bus takes T minutes: centred on 12, with variance 4.
$P (8 < T < 16) \approx 0.95$ That is a standard deviation of 2, so about 95 walks in 100 take between 8 and 16 minutes.
$P (R) P (U) = 0.12 \neq = 0.27 = P (R \cap U)$ 0.3 times 0.4 is 0.12, not 0.27. Rain and umbrellas are not independent: seeing one is a clue about the other.

In words this story is 787 letters long; in symbols, 164.

Read it

What does it say?

$P (A ∣ B) = 0.8$

$E [X] = 3$

$X \sim N (0, 1)$

$A ⊥ B$

$P (A) = 0$

$P (A \cap B) = 0.1$

$P (B ∣ A) = 1$

$Var (X) = 0$

$E [X] = 2.5$

$X \sim N (70, 25)$

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

$P (A ∣ B) = P (A)$

Write it

Which notation says this?

The chance of A once you know B has happened.

The variance of X is the average squared distance from the mean.

X is a random value following the bell curve with mean μ and variance σ².

The chance that both A and B happen.

The chance of B once you know A has happened.

The average of X over very many repeats is 4.

A and B are independent.

X follows the bell curve centred on 50 with standard deviation 10.

The chance of A given B is the chance of both, divided by the chance of B.

The chance that it rains tomorrow, R, is 0.3.

The variance of X is 9.

The variance of X is the average of the squares minus the square of the average.

Build it

Tap the tiles in order. One tile does not belong.

The chance of A and B together is the chance of A times the chance of B.

The chance of R given U is the chance of R and U together, divided by the chance of U.

The chance of A given B is 0.9.

X follows the bell curve with mean 12 and variance 4.

The expectation of X is the sum over k of k times the chance that X equals k.

The variance of X is the expectation of X minus mu, squared.

Challenges

Creative puzzles that use every sign of this chapter. Take your time; there is no clock.

1 One die, three events

A fair die is rolled; each face has chance 1/6. A tick shows which faces belong to each event: A, B, C. Judge each sentence.

	$A$	$B$	$C$
$1$		✓
$2$	✓	✓
$3$		✓
$4$	✓	✓	✓
$5$			✓
$6$	✓		✓

A tick means the face (row) belongs to the event (column).

$P (A) = \frac{1}{2}$
$A ⊥ B$
$A ⊥ C$
$P (B ∣ A) = \frac{2}{3}$
$P (A ∣ B) = P (B ∣ A)$
$P (A \cap C) = \frac{1}{3}$

2 Given, together, average

Pair each line with what it says.

Tap a line, then its meaning.

3 Spread from averages

This should say: the variance is the average of the squares, minus the square of the average. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

4 Narrowing down

This should say: the chance of A given B is the chance of both, divided by the chance of what we now know. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

5 The fair die

What is the average of one roll of a fair die?

$E [X] = 1 \cdot \frac{1}{6} + 2 \cdot \frac{1}{6} + \dots + 6 \cdot \frac{1}{6}$

6 Two independent chances

Two independent events: what is the chance that both happen?

$P (A) = 0.5, P (B) = 0.4, A ⊥ B ∴ P (A \cap B) = ?$

7 Reading the bell

X follows a bell curve centred on 10, with the second number in N(·, ·) as the variance. What is the variance of X?

$X \sim N (10, 9) ∴ Var (X) = ?$

8 The test and the illness

S: the person is ill. T: the test is positive. A test is positive far more often for the ill than the ill are among all positives. Write that “ill given positive” and “positive given ill” are different, then compare.

Your version

9 Six days, rain and umbrellas

Six days; a day is picked at random. R: it rained. U: the umbrella came along. C: it was cold. Judge each sentence.

	$R$	$U$	$C$
$d_{1}$	✓	✓	✓
$d_{2}$	✓	✓
$d_{3}$	✓	✓
$d_{4}$		✓	✓
$d_{5}$
$d_{6}$

A tick means the event (column) happened on that day (row). Each day has chance 1/6.

$P (U ∣ R) = 1$
$P (R ∣ U) = 1$
$P (R ∣ U) = \frac{3}{4}$
$R ⊥ C$
$U ⊥ C$
$P (R \cap U) = P (R) P (U)$

10 Two coins

A fair coin is tossed twice; the four outcomes are equally likely. A: the first toss is heads. B: at least one tails. C: both tosses agree. Judge each sentence.

	$A$	$B$	$C$
$H H$	✓		✓
$H T$	✓	✓
$T H$		✓
$T T$		✓	✓

A tick means the outcome (row) belongs to the event (column). H is heads, T is tails; each outcome has chance 1/4.

$P (B) = \frac{3}{4}$
$A ⊥ C$
$P (A ∣ B) = \frac{1}{3}$
$P (B ∣ A) = \frac{1}{3}$
$A ⊥ B$
$P (A \cap C) = \frac{1}{4}$

11 Five numbers about X

Five things one can say about a random X. Pair each with its meaning.

Tap a line, then its meaning.

12 The second number in the bell

This should say: X follows the bell curve with mean μ and variance σ². One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

13 Independent means multiply

This should define independence: the chance of both is the product of the chances. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

14 Two dice

Two fair dice are rolled and the faces are added. What is the average total?

$E [X_{1}] = E [X_{2}] = 3.5, E [X_{1} + X_{2}] = ?$

15 One coin

X is 1 for heads and 0 for tails. How spread out is X?

$P (X = 1) = P (X = 0) = \frac{1}{2}, Var (X) = ?$

16 From both to given

Half the days are windy (A); on a fifth of all days it is windy and the kite flies (B). Given wind, what is the chance the kite flies?

$P (A) = 0.5, P (A \cap B) = 0.2, P (B ∣ A) = ?$

17 Within one sigma

Scores follow a bell curve centred on 100 with variance 225. About what share of scores lie between 85 and 115?

$X \sim N (100, 225), P (85 < X < 115) \approx ?$

18 Clouds in the morning

R: it rains today. C: the morning is cloudy. Write that a cloudy morning makes rain more likely than usual, then compare.

Your version

19 The coin has no memory

H₁: the first toss is heads. H₂: the second is heads. Write that the two tosses do not influence each other, and what the chance of two heads is, then compare.

Your version

Tell your own

Tell a two-line story in notation about a chance that changes when you learn something: a bus, a match, a bag of sweets. Use P, and “given”.

Kept on this device only.

Probability: how likely, and given what

The alphabet

The story: Rain and umbrellas

Read it

Write it

Build it

Challenges

1 One die, three events

2 Given, together, average

3 Spread from averages

4 Narrowing down

5 The fair die

6 Two independent chances

7 Reading the bell

8 The test and the illness

9 Six days, rain and umbrellas

10 Two coins

11 Five numbers about X

12 The second number in the bell

13 Independent means multiply

14 Two dice

15 One coin

16 From both to given

17 Within one sigma

18 Clouds in the morning

19 The coin has no memory

Tell your own

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