Probability & Statistics 3D Lab

Unit 1

Probability is counting inside a sample space

Every probability in this unit is a fraction: favourable outcomes over all outcomes. Conditional probability shrinks the sample space. Bayes' theorem works backwards from an effect to its cause.

Download Unit 1 notes (PDF) 17 pages

Formulas to remember

P(E) = n(E)/n(S)
P(A∪B) = P(A)+P(B)−P(A∩B)
P(B|A) = P(A∩B)/P(A)
P(E_k|D) = P(E_k)P(D|E_k) / ΣP(E_i)P(D|E_i)

1.1Two dice: the 36-cube sample space

Each cube is one outcome (die 1, die 2). Choose an event. The favourable cubes rise. The probability is the number of raised cubes divided by 36.

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Event

k7

event Aevent B / givenA ∩ B

What you see

The floor is the sample space S, so n(S) = 36. A raised cube belongs to the event.

Addition theorem

In the OR mode, purple cubes are counted in both A and B. That is why we subtract P(A∩B) once.

Conditional

In the GIVEN mode, only the blue and purple cubes remain. They form the new sample space. Count inside them only.

1.2Bayes' theorem as blocks of output

Each block is one source (machine, hotel, person). Its width is the prior P(E_i). The red part at the bottom is P(D|E_i). The back row shows the posteriors: each red share divided by all the red.

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Prior A (%)

Prior B (%)

Prior C (%)= 100 − A − B

P(D|A) %

P(D|B) %

P(D|C) %

Total probability

All the red together is P(D) = Σ P(E_i)P(D|E_i).

Bayes

The posterior of a source is its red share of the total red. A small factory with a high defect rate can own most of the red.

Exam check

The posteriors in the back row always add to 100%. Use this to check your answer.

Unit 2

Distributions are shapes made of probability

A discrete distribution is a set of bars that add to 1. A continuous one is a curve with area 1. The Galton board shows how many small coin flips build the normal curve.

Download Unit 2 notes (PDF) 19 pages

Formulas to remember

Binomial: ⁿC_rp^rq^n−r, μ=np, σ²=npq
Poisson: e^−λλ^r/r!, μ=σ²=λ
Normal: Z=(X−μ)/σ
S.E. of x̄ = σ/√n

2.1Binomial and Poisson bars

Each bar is P(X = r). Pick r to see one bar (orange) and the cumulative P(X ≤ r) (light orange). The red line is the mean. The blue band is mean ± σ.

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Distribution show Poisson(λ = np) behind

n (trials)

p (success)

r

Shape

Binomial is symmetric only when p = 0.5. Poisson is skewed right for small λ.

Binomial → Poisson

Set n = 40 and p small, then switch on the Poisson bars. They almost match, because λ = np.

Mean vs variance

Binomial: variance npq is less than the mean np. Poisson: variance equals the mean.

2.2Galton board: Binomial becomes Normal

Each ball hits 12 pegs. At each peg it goes right with probability p. Its final bin is the number of rights, which is a Binomial(12, p) count. Drop many balls and the bins take the shape of the normal curve (orange line).

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p (go right)

One peg = one trial

Right is "success". 12 pegs means n = 12 independent trials with the same p.

Central limit idea

A sum of many small random steps is approximately normal. Mean np, s.d. √(npq).

Try it

Set p = 0.2. The pile shifts left (mean = 2.4) and becomes skewed.

2.3Normal curve: probability is area

Enter μ, σ and the limits. The orange slab is the probability. The axis shows both the real values and the z-values, so you can see the change Z = (X − μ)/σ.

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μ σ N

a a = −∞

b b = +∞

Same side

Both z-values on one side of 0: subtract the areas, A(z₂) − A(z₁).

Opposite sides

One z negative, one positive: add the areas, A(z₁) + A(z₂).

Tails

Beyond one value: 0.5 − A(z). "How many" means multiply the probability by N.

Unit 3

A test asks: is Z inside the red region?

Every large-sample test computes Z = (statistic − parameter) / S.E. If Z lands in the critical region, we reject H₀. The confidence-interval scene shows what "95% confident" really means.

Download Unit 3 notes (PDF) 13 pages

Critical values

two-tailed: 1.96 (5%), 2.58 (1%), 2.33 (2%)
one-tailed: 1.645 (5%), 2.33 (1%)
CI: statistic ± z × S.E.
n = (zσ/E)²

3.1The Z-test machine

Choose the test and enter the numbers. The pin shows your Z. The red regions are the critical region for the chosen α and tail. The panel writes the six exam steps for you.

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Test

H₁ α

Acceptance region

The blue middle part. If Z is here, the difference is due to sampling chance. Accept H₀.

Critical region

The red tail area equals α. If Z is here, the difference is too large to be chance. Reject H₀.

Read H₁ from words

"Superior, better, more" → right-tailed. "Less, at least is doubted" → left-tailed. "Same, differ" → two-tailed.

3.2Type I and Type II errors

The front curve is the distribution of Z when H₀ is true. The back curve is the distribution when H₁ is true. Red is α (reject a true H₀). Grey is β (accept a false H₀).

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α

true shift δ

n multiplier

Trade-off

Make α smaller and the cut moves right. Then β grows. For a fixed n you cannot make both small.

Bigger sample

Increase n. The curves separate (δ grows with √n), so β falls with the same α.

Names

Type I = producer's risk (good lot rejected). Type II = consumer's risk (bad lot accepted). Power = 1 − β.

3.3What "95% confidence" means

40 samples are drawn from a population with true mean μ = 50 and σ = 10. Each bar is one interval x̄ ± z·σ/√n. The glass wall is μ. Red bars miss μ.

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Confidence

sample size n

Meaning

About 95 of every 100 intervals contain μ. One single interval either contains μ or not.

Width

Width = 2z·σ/√n. Higher confidence → wider bars. Larger n → narrower bars.

Max error

Half the width is the maximum error E = z·σ/√n. So n = (zσ/E)².

Unit 4

Small samples need wider critical values

With n < 30 we estimate σ from the sample, so the t-curve has thicker tails than Z. χ² compares observed with expected counts. F compares two variances.

Download Unit 4 notes (PDF) 15 pages

Formulas to remember

t = (x̄−μ)/(S/√n), ν=n−1
two means: ν=n₁+n₂−2 · paired: t=d̄/(S_d/√n)
F = S₁²/S₂² (larger on top)
χ² = Σ(O−E)²/E

4.1The t-family as a landscape

Each slice of this surface is a t-curve for one value of ν (degrees of freedom). The orange slice is your ν. The blue curve in front is the standard normal. The red pins are ±t_α/2.

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ν = n − 1

α (two-tailed)

your t

Why thicker tails

S changes from sample to sample. That extra uncertainty spreads the curve. So |t| must be larger before we reject.

Large ν

Move ν to 30. The slice almost matches the normal curve and t_0.025 → 1.96.

Exam rule

|t| < t_tab(ν): accept H₀. Always write ν in your answer.

4.2χ² goodness of fit: observed vs expected

Front row: observed O (orange). Middle row: expected E (blue glass). Back row: each cell's share (O − E)²/E (purple). χ² is the total of the back row.

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Data

push data

Big χ²

Large gaps between O and E make tall purple bars. If χ² is more than the table value, reject the fit.

Degrees of freedom

ν = classes − 1. Subtract one more for each parameter you estimate (λ for Poisson).

Try it

Move "push data". The observed bars drift away from E and χ² crosses the critical value.

Unit 5

Least squares makes the error squares smallest

Each error e = y − f(x) is drawn as a real square. The best-fit curve is the one with the smallest total square area. Logs turn curves into straight lines.

Download Unit 5 notes (PDF) 7 pages

Normal equations

line: na+bΣx=Σy ; aΣx+bΣx²=Σxy
parabola: multiply by 1, x, x² and add
ae^bx: Y=log y, b = B/0.4343
ax^b: X=log x, Y=log y

5.1Error squares you can see

Orange squares belong to the least-squares curve. Turn on "my curve" and move a, b, c. Your blue squares grow, and your S is always larger than the best S.

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Data Fit

my curve

a

b

c

Why squares

Positive and negative errors would cancel. Squares are always positive, so S = Σe² measures the total misfit.

How the minimum is found

Set ∂S/∂a = 0 and ∂S/∂b = 0. These give the normal equations shown in the panel.

Class-note check

Load "cars inspected" with a parabola. The best curve is y = 10.822 + 1.717x − 0.065x², not the class-note curve.

5.2Logs straighten a curve

Move the slider from "original" to "log scale". The points and the fitted curve bend into a straight line. That is why we fit Y = A + Bx after taking logs.

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Data

original → log

ae^bx

log y = log a + (b log e)x. Only y is logged. So B = 0.4343b and b = B/0.4343.

ax^b

log y = log a + b log x. Both axes are logged. The slope is b directly.

ab^x

log y = log a + x log b. Then b = 10^B. It is the same curve as ae^bx.

Unit 6

Correlation tilts the cloud, regression draws two lines

r measures how tightly the points follow a line. There are two regression lines: y on x and x on y. They cross at the means, and the angle between them closes as |r| → 1.

Download Unit 6 notes (PDF) 10 pages

Formulas to remember

r = ΣXY / √(ΣX²ΣY²)
b_yx = rσ_y/σ_x, b_xy = rσ_x/σ_y
r² = b_xy·b_yx
ρ = 1 − 6[ΣD² + Σ(m³−m)/12] / n(n²−1)

6.1The correlation hill

The hill is the joint density of x and y. The dots on the floor are a sample. Blue is the line of y on x. Orange is the line of x on y. Both pass through (x̄, ȳ).

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r

σx

σy

r = ±1

The hill becomes a thin ridge, the two lines meet, and the angle is 0.

r = 0

The hill is round. The lines are perpendicular (horizontal and vertical).

Sign rule

b_yx, b_xy and r always have the same sign. If b_xy·b_yx > 1, you labelled the lines wrongly.

6.2Spearman's rank correlation as threads

Front row: ranks in x. Back row: ranks in y. A thread joins each student's two ranks. Straight threads (green) mean agreement. Long slanted threads (red) give large D².

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Data

Ties

Equal values share the average rank (e.g. 3.5, 3.5). Stacked spheres show the ties. Add (m³ − m)/12 for each tie group.

Agreement

All threads straight → ΣD² = 0 → ρ = 1. Ranks fully reversed → ρ = −1.

Exam check

ΣD always equals 0. Check this before you square.

Probability is counting inside a sample space

What you see

Addition theorem

Conditional

Total probability

Bayes

Exam check

Distributions are shapes made of probability

Shape

Binomial → Poisson

Mean vs variance

One peg = one trial

Central limit idea

Try it

Same side

Opposite sides

Tails

A test asks: is Z inside the red region?

Acceptance region

Critical region

Read H₁ from words

Trade-off

Bigger sample

Names

Meaning

Width

Max error

Small samples need wider critical values

Why thicker tails

Large ν

Exam rule

Big χ²

Degrees of freedom

Try it

Least squares makes the error squares smallest

Why squares

How the minimum is found

Class-note check

aebx

axb

abx

Correlation tilts the cloud, regression draws two lines

r = ±1

r = 0

Sign rule

Ties

Agreement

Exam check

ae^bx

ax^b

ab^x