IB Math AA SL Topic 4.8: Binomial distribution | Notes & Flashcards | Aimnova

IB Math AA SL — Binomial distribution

Topic 4.8 of IB Mathematics: Analysis and Approaches covers Binomial distribution, which is part of Unit 4: Statistics & Probability. Students explore key concepts including Binomial probabilities, Mean & variance. A strong understanding of binomial distribution is essential for IB Math AA SL exams and builds the foundation for connected topics across the syllabus.

Key concepts in Binomial distribution

Key Idea: The binomial distribution counts successes in a fixed number of independent trials — and it is a heavy Paper 2 GDC topic, with binompdf / binomcdf doing nearly all the work.

🎯 When is it binomial? — X ~ B(n, p)

Fixed number of trials n, each trial independent, only two outcomes (success / failure), and the same success probability p every time. Then X ~ B(n, p) counts the successes. Drawing without replacement changes p between trials → not binomial.

🧮 Probabilities on the GDC (Paper 2)

P(X = k) = \binom{n}{k}\,p^{k}(1-p)^{n-k}

$n$: number of trials
$p$: success probability
$k$: number of successes

📊 Mean & variance

E(X) = np, \qquad \text{Var}(X) = np(1-p), \qquad \sigma = \sqrt{np(1-p)}

$np$: expected number of successes
$np(1-p)$: variance — multiply by both p and (1 − p)

Tip: Given the mean and variance, variance ÷ mean = 1 − p (this cancels n) → gives p; then n = mean ÷ p.

✏️ IB-style worked examples

IB-style question — exactly, at most, at least

A spinner lands on red with probability 0.35. It is spun 14 times. Let X be the number of reds. Find (a) P(X = 5), (b) P(X ≤ 4), (c) P(X ≥ 6).

Step by step:

State the model.
$X \sim B(14,\,0.35)$
(a) Exactly 5 → binompdf(14, 0.35, 5).
$P(X = 5) \approx 0.218$
(b) At most 4 → binomcdf(14, 0.35, 4).
$P(X \le 4) \approx 0.423$
(c) At least 6 is the complement of ≤ 5.
$P(X \ge 6) = 1 - \text{binomcdf}(14, 0.35, 5) \approx 0.359$

Final answer:

(a) 0.218 (b) 0.423 (c) 0.359 (3 s.f.).

IB-style question — find the mean, variance and standard deviation

A quiz has 40 independent questions, each guessed correctly with probability 0.25. Let X be the number correct. Find the mean, variance and standard deviation of X.

Step by step:

Mean = np.
$E(X) = 40 \times 0.25 = 10$
Variance = np(1 − p).
$\text{Var}(X) = 40 \times 0.25 \times 0.75 = 7.5$
Standard deviation = √variance.
$\sigma = \sqrt{7.5} \approx 2.74$

Final answer:

Mean = 10, variance = 7.5, standard deviation ≈ 2.74.

Important: binomcdf only gives P(X ≤ k), so an 'at least k' or 'more than k' question needs the complement: P(X ≥ k) = 1 − binomcdf(n, p, k − 1) and P(X > k) = 1 − binomcdf(n, p, k). Forgetting the 1 −, or the k − 1, is the most common lost mark in this topic.

Tap each card to reveal the answer.

Exam Tips

Always state the model first: X ~ B(n, p), then read off n, p and k.
binompdf = exactly k; binomcdf = at most k (≤). They live in DISTR (2nd → VARS).
'At least / more than' → use the complement: P(X ≥ k) = 1 − binomcdf(n, p, k − 1).
Mean = np, variance = np(1 − p), sd = √(np(1 − p)) — all in the formula booklet.
To find n and p from mean & variance: variance ÷ mean = 1 − p, then n = mean ÷ p.

IB Math AA SL — Binomial distribution

Key concepts in Binomial distribution

Key Idea: The binomial distribution counts successes in a fixed number of independent trials — and it is a heavy Paper 2 GDC topic, with binompdf / binomcdf doing nearly all the work.

🎯 When is it binomial? — X ~ B(n, p)

Fixed number of trials n, each trial independent, only two outcomes (success / failure), and the same success probability p every time. Then X ~ B(n, p) counts the successes. Drawing without replacement changes p between trials → not binomial.

🧮 Probabilities on the GDC (Paper 2)

P(X = k) = \binom{n}{k}\,p^{k}(1-p)^{n-k}

$n$: number of trials
$p$: success probability
$k$: number of successes

📊 Mean & variance

E(X) = np, \qquad \text{Var}(X) = np(1-p), \qquad \sigma = \sqrt{np(1-p)}

$np$: expected number of successes
$np(1-p)$: variance — multiply by both p and (1 − p)

Tip: Given the mean and variance, variance ÷ mean = 1 − p (this cancels n) → gives p; then n = mean ÷ p.

✏️ IB-style worked examples

IB-style question — exactly, at most, at least

A spinner lands on red with probability 0.35. It is spun 14 times. Let X be the number of reds. Find (a) P(X = 5), (b) P(X ≤ 4), (c) P(X ≥ 6).

Step by step:

State the model.
$X \sim B(14,\,0.35)$
(a) Exactly 5 → binompdf(14, 0.35, 5).
$P(X = 5) \approx 0.218$
(b) At most 4 → binomcdf(14, 0.35, 4).
$P(X \le 4) \approx 0.423$
(c) At least 6 is the complement of ≤ 5.
$P(X \ge 6) = 1 - \text{binomcdf}(14, 0.35, 5) \approx 0.359$

Final answer:

(a) 0.218 (b) 0.423 (c) 0.359 (3 s.f.).

IB-style question — find the mean, variance and standard deviation

A quiz has 40 independent questions, each guessed correctly with probability 0.25. Let X be the number correct. Find the mean, variance and standard deviation of X.

Step by step:

Mean = np.
$E(X) = 40 \times 0.25 = 10$
Variance = np(1 − p).
$\text{Var}(X) = 40 \times 0.25 \times 0.75 = 7.5$
Standard deviation = √variance.
$\sigma = \sqrt{7.5} \approx 2.74$

Final answer:

Mean = 10, variance = 7.5, standard deviation ≈ 2.74.

Important: binomcdf only gives P(X ≤ k), so an 'at least k' or 'more than k' question needs the complement: P(X ≥ k) = 1 − binomcdf(n, p, k − 1) and P(X > k) = 1 − binomcdf(n, p, k). Forgetting the 1 −, or the k − 1, is the most common lost mark in this topic.

Tap each card to reveal the answer.

Exam Tips

Always state the model first: X ~ B(n, p), then read off n, p and k.
binompdf = exactly k; binomcdf = at most k (≤). They live in DISTR (2nd → VARS).
'At least / more than' → use the complement: P(X ≥ k) = 1 − binomcdf(n, p, k − 1).
Mean = np, variance = np(1 − p), sd = √(np(1 − p)) — all in the formula booklet.
To find n and p from mean & variance: variance ÷ mean = 1 − p, then n = mean ÷ p.

IB Math AA SL — Binomial distribution

Key concepts in Binomial distribution

🎯 When is it binomial? — X ~ B(n, p)

🧮 Probabilities on the GDC (Paper 2)

📊 Mean & variance

✏️ IB-style worked examples

IB-style question — exactly, at most, at least

IB-style question — find the mean, variance and standard deviation

Exam Tips

What you'll learn in Topic 4.8

Study resources — 4.8 Binomial distribution

Binomial probabilities

Mean & variance

Ready to study Binomial distribution?

Ready to practice?

IB Math AA SL — Binomial distribution

Key concepts in Binomial distribution

🎯 When is it binomial? — X ~ B(n, p)

🧮 Probabilities on the GDC (Paper 2)

📊 Mean & variance

✏️ IB-style worked examples

IB-style question — exactly, at most, at least

IB-style question — find the mean, variance and standard deviation

Exam Tips

What you'll learn in Topic 4.8

Study resources — 4.8 Binomial distribution

Binomial probabilities

Mean & variance

Ready to study Binomial distribution?

Ready to practice?