IB Math AA HL Probability Distributions — Complete Cheatsheet

Probability Distributions is the largest single topic in IB Mathematics Analysis & Approaches HL — a single Section B question can carry 14 to 22 marks, and combined Binomial-Normal questions appear on almost every Paper 2. The traps are silent: writing $N(\mu, \sigma)$ instead of $N(\mu, \sigma^2)$, multiplying instead of squaring in $\text{Var}(aX + b)$, rounding the intermediate $p$ between sub-parts, or forgetting that "at least 1" should be solved by complement. The GDC saves you nothing if your distribution statement is wrong.

This cheatsheet condenses every formula and GDC trick from SL 4.7–4.9 and AHL 4.14–4.18 into one revisable page. It is organised into three major blocks — a general intro to discrete and continuous random variables, a full treatment of the Binomial distribution, and a full treatment of the Normal distribution — and finishes with the combined Binomial-Normal exam pattern that defines HL Section B. Scroll to the bottom for the printable PDF and gated full library.

§1 — Discrete & Continuous Random Variables SL 4.7–4.8

Discrete random variable

Continuous random variable (PDF)

Discrete vs Continuous

§2 — Binomial Distribution: BINS & Notation SL 4.8, AHL 4.14

BINS checklist

• B — Fixed $n$ trials.
• I — Independent trials.
• N — 2 outcomes only (success/failure).
• S — Same $p$ each trial.

All four conditions must hold for the binomial model to apply.

Notation

• $X \sim B(n, p)$.
• $n$ = number of trials.
• $p$ = probability of success.
• $X$ = number of successes; $X = 0, 1, 2, \ldots, n$.

§3 — Binomial PDF, Mean & Variance SL 4.8, AHL 4.14

PDF formula

$$P(X = r) = \binom{n}{r} p^r (1 - p)^{n-r} = \dfrac{n!}{r!(n - r)!}\, p^r q^{n-r}, \quad q = 1 - p$$

Mean and variance

§4 — GDC & CDF Translation SL 4.8

GDC syntax

• TI-84 exact: binompdf(n,p,r) $= P(X = r)$.
• TI-84 cumulative: binomcdf(n,p,r) $= P(X \le r)$.
• $P(X \ge r)$: 1 - binomcdf(n,p,r-1).
• Casio: STAT $\to$ DIST $\to$ BINM. Bpd: exact. Bcd: cumulative.

CDF phrases — translation table

Conditional probability for binomial

$$P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$$

Typical: $P(X = r \mid X \le k) = \dfrac{P(X = r)}{P(X \le k)}$ — only valid when $r \le k$.

§5 — Finding Minimum $n$ AHL 4.14

Method 1 — algebraic

1. Translate "$P(X \ge 1) > k$" to "$P(X = 0) < 1 - k$".
2. Substitute: $(1 - p)^n < 1 - k$.
3. Take logs (note both sides negative): $n > \dfrac{\ln(1 - k)}{\ln(1 - p)}$.
4. Round UP to the next integer. Verify by substituting back.

Method 2 — GDC table/graph

1. Set $Y_1 = $ binomcdf(X, p, r), $Y_2 = $ target.
2. GRAPH $\to$ CALC $\to$ Intersect, OR use TABLE with $\Delta\text{Tbl} = 1$ and scroll.
3. Always verify by substituting back.

§6 — Linear Transformations & Finding $p$ AHL 4.14

Linear transformations $Y = aX + b$

Adding $b$ shifts the expectation but does not change the variance. The coefficient $a$ is always squared, even if $a < 0$. Example: $Y = 2 - 5X$, so $a = -5$, $b = 2$. $E(Y) = 2 - 5 E(X)$ and $\text{Var}(Y) = (-5)^2 \text{Var}(X) = 25 \text{Var}(X)$.

Finding $p$ from mean / variance

Given $E(X) = np = m$ and $\text{Var}(X) = np(1 - p)$:

$$1 - p = \dfrac{\text{Var}(X)}{E(X)} = \dfrac{np(1 - p)}{np}, \quad \text{then } n = \dfrac{E(X)}{p}$$

§7 — False Binomials & Combined Patterns AHL 4.14–4.18

False binomials

• First success: Geometric distribution, $P(X = k) = (1 - p)^{k - 1} p$.
• Without replacement, small $N$: Hypergeometric (not in syllabus).
• Dependent trials: Neither binomial nor geometric.
• Exception: if $N$ is large and you sample without replacement, the distribution is approximately Binomial.

Normal $\to$ Binomial pattern

1. Stage 1: use the Normal distribution to find $p = P(\text{item meets condition})$.
2. Stage 2: a sample of $n$ items, $X \sim B(n, p)$.
3. Trigger phrase: "A sample of $n$ is selected...".
4. Carry the exact GDC value for $p$ — never round between parts.

§8 — Normal Distribution Properties SL 4.9, AHL 4.14

$X \sim N(\mu, \sigma^2)$. Key properties:

• Symmetric about $\mu$.
• Mean $=$ Median $=$ Mode $= \mu$.
• $P(X = k) = 0$ always (continuous).
• Total area under the curve $= 1$.
• Inflection points at $\mu \pm \sigma$.
• Asymptotic to the $x$-axis.
• $\sigma \uparrow$: wider; $\sigma \downarrow$: narrower.

§9 — Empirical Rule (P1, no GDC) SL 4.9

The 68/95/99.7 rule — MEMORISE

• $P(\mu - \sigma < X < \mu + \sigma) \approx 68\%$
• $P(\mu - 2\sigma < X < \mu + 2\sigma) \approx 95\%$
• $P(\mu - 3\sigma < X < \mu + 3\sigma) \approx 99.7\%$

Derived values (one-tail):

• $P(X > \mu + \sigma) \approx 16\%$
• $P(X > \mu + 2\sigma) \approx 2.5\%$
• $P(X > \mu + 3\sigma) \approx 0.15\%$
• $P(\mu < X < \mu + \sigma) \approx 34\%$
• $P(\mu + \sigma < X < \mu + 2\sigma) \approx 13.5\%$

Symmetry rules (Paper 1, no GDC)

• $P(X < \mu) = 0.5$.
• $P(X < \mu - k) = P(X > \mu + k)$.
• $P(\mu < X < \mu + k) = P(\mu - k < X < \mu)$.
• Same $\sigma$, different $\mu$: $b - \mu_X = c - \mu_Y \Rightarrow P(X > b) = P(Y > c)$.

§10 — Standardisation & $z$-scores AHL 4.14

If $X \sim N(\mu, \sigma^2)$, then:

$$Z = \dfrac{X - \mu}{\sigma} \sim N(0, 1)$$

• Interpretation 1: $z$ = number of standard deviations above the mean. $z > 0$: above; $z < 0$: below.
• Interpretation 2: $z$ = position on the standard scale (mean 0, SD 1).
• "$k$ SDs above mean" $\Rightarrow z = k \Rightarrow X = \mu + k\sigma$.

§11 — Normal GDC Skills & Finding $\mu, \sigma$ SL 4.9, AHL 4.14

GDC normal syntax

• TI-84 forward: normalcdf(a, b, mu, sigma). Use $\pm 10^{99}$ for $\pm\infty$.
• TI-84 inverse: invNorm(p, mu, sigma) gives $x$ such that $P(X < x) = p$.
• For $P(X > x) = p$: use $1 - p$ in invNorm.
• Casio: STAT $\to$ DIST $\to$ NORM. Ncd: forward. InvN: inverse. Casio: enter $\sigma$ before $\mu$!

Finding $\mu$ and $\sigma$

1. Step 1: from $P(X < a) = p$, get $z = $ invNorm(p, 0, 1).
2. Step 2: use $\dfrac{a - \mu}{\sigma} = z$.
3. Two unknowns: two conditions $\to$ two equations, solve simultaneously.
4. IQR method: $\sigma = \dfrac{\text{IQR}}{2 \times 0.6745}$   ($z_{0.75} = 0.6745$ — memorise).

§12 — Two-Stage & Comparison Patterns AHL 4.14–4.18

Two-stage pattern: Normal $\to$ Binomial

1. Find $p = P(\text{condition})$ from the Normal distribution.
2. Sample of $n$ items: $X \sim B(n, p)$.
3. Trigger: "a sample of $n$...".
4. Carry the exact GDC $p$ forward — never round between parts.

Combined exam pattern (typical Section B)

• Part (a): basic normal probability $\to$ Part (b): find $\sigma$ via invNorm.
• Part (c): use that $p$ in Binomial $\to$ Part (d): conditional probability or find $n$.

Comparing two distributions

Given $X \sim N(\mu_1, \sigma^2)$ and $Y \sim N(\mu_2, \sigma^2)$:

$$P(X > a) = P(Y > b) \iff \dfrac{a - \mu_1}{\sigma} = \dfrac{b - \mu_2}{\sigma} \iff a - \mu_1 = b - \mu_2$$

Percentile comparison: compute $z_X = \dfrac{x - \mu_X}{\sigma_X}$ and $z_Y = \dfrac{y - \mu_Y}{\sigma_Y}$. Larger $z$ means relatively better performance.

§13 — Exam Traps & Attack Plan All sections

Top exam traps

1. $N(\mu, \sigma^2)$: write $\sigma^2$, not $\sigma$.
2. $\text{Var}(Y = aX + b) = a^2 \text{Var}$, not $a \cdot \text{Var}$.
3. Both values of $p$ when solving the variance quadratic.
4. Round UP, not down, for minimum $n$.
5. "First success" is geometric, not binomial: $(1 - p)^{k - 1} p$.
6. "At least 1" $= 1 - P(X = 0)$.
7. Casio: enter $\sigma$ before $\mu$.
8. Without replacement: check whether $N$ is small.
9. The 68/95/99.7 rule is exact only at $\pm 1, 2, 3 \sigma$.
10. State the distribution before the GDC step — earns M1.

Attack plan — when you see, do this

Notation summary

• $X \sim B(n, p)$ — Binomial.
• $X \sim N(\mu, \sigma^2)$ — Normal.
• $Z \sim N(0, 1)$ — Standard Normal.
• $E(X) = \mu$, $\text{Var}(X) = \sigma^2$.

Always write the distribution first, then use the GDC — this earns M1 every time.

Worked Example — Combined Normal & Binomial

Common Student Questions