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Topic 4.13 · HL only

Bayes theorem: notes and practice questions

3 min read Jump to 3 practice questions
Summary
  • Binomial Distribution: Models the number of successes in a fixed number of independent trials with constant success probability (nn, pp). Key stats: Mean = npnp, Variance = np(1−p)np(1-p). Use Binomial PDF for exact probabilities and CDF for cumulative ones.
  • Normal Distribution: A continuous distribution, denoted X∽N(μ,σ2)X \backsim N(\mu, \sigma^2). Characterized by its bell curve. Standardize using Z=(X−μ)/σZ = (X - \mu) / \sigma to get Z∽N(0,1)Z \backsim N(0, 1). Use Normal CDF to find probabilities (areas) and Inverse Normal to find values given probabilities. Empirical rule: ~68% within μ±σ\mu \pm \sigma, ~95% within μ±2σ\mu \pm 2\sigma, ~99.7% within μ±3σ\mu \pm 3\sigma.

How it is examined

Three events is the ceiling, so a question with a four-branch partition is out of syllabus. The medical-testing context (false positives) is the archetype. A tree diagram is usually the faster route and is fully acceptable, since the guidance never demands the formula. 5 to 7 marks.

Given in the booklet

Bayes' theorem is given for two and for three events.

Key ideas

Use of Bayes' theorem for a maximum of three events.

Linking questions

  • Other contexts: probability methods in medical studies to assess risk factors for certain diseases.

Practice questions

3 questions · 1 medium · 2 hard
Showing 3 of 3

Question 1

MediumPaper 1 · no calculator7 marks
(a)

A factory produces microchips on two production lines, Line 1 and Line 2. Line 1 produces 60%60\% of the microchips and Line 2 produces the remaining 40%40\%. The probability that a microchip from Line 1 is defective is 0.050.05, and the probability that a microchip from Line 2 is defective is 0.080.08.

A microchip is selected at random from the factory's total output.

(a) Find the probability that it is defective.

[3]
(b)

(b) Given that a randomly selected microchip is defective, find the probability that it was produced by Line 1.

[2]
(c)

(c) find the probability that it was produced by Line 2.

[2]

Question 2

HardPaper 2 · calculator5 marks
(a)

A factory produces two types of light bulbs: standard and long-life. The lifespan of the bulbs, in hours, can be modelled as normal distributions with the following parameters.

Bulb typeMean μ\muStandard deviation σ\sigma
Standard800 h50 h
Long-life1000 h100 h

(a) Find the percentage of standard bulbs that have a lifespan of less than 700 hours.

[1]
(b)

(b) The factory produces a large number of bulbs, of which 60% are standard bulbs. Both types of bulbs are produced and randomly mixed together for packaging. A quality control process identifies and removes all bulbs with a lifespan of less than 700 hours. An inspector randomly selects a bulb from this removed group. Find the probability that it is a standard bulb.

[4]

Question 3

HardPaper 2 · calculator16 marks
(a)

(a) An electronics factory produces two types of resistors: Type A and Type B.

The resistance, RAR_A (in Ohms), of Type A resistors is normally distributed with a mean of 120 Ohms and a standard deviation of 5 Ohms.

Find the probability that a randomly selected Type A resistor has a resistance less than 115 Ohms.

[2]
(b)

(b) In a random selection of 10 Type A resistors, find the probability that exactly 3 have a resistance less than 115 Ohms.

[2]
(c)(i)

(c) The resistance, RBR_B (in Ohms), of Type B resistors is normally distributed with a mean of 135 Ohms and a standard deviation of 7 Ohms.

Each day, 70% of the resistors produced are Type A, and 30% are Type B.

On a particular day, a resistor is randomly selected from all those produced at the factory.

Let AA represent 'Type A resistor' and BB represent 'Type B resistor'.

(i) Find the probability that the randomly selected resistor has a resistance less than 115 Ohms.

[4]
(c)(ii)

(ii) Given that a randomly selected resistor has a resistance less than 115 Ohms, find the probability that it is a Type A resistor.

[3]
(d)

(d) The machine that makes the Type A resistors is adjusted so that the mean resistance of the Type A resistors remains the same, but their standard deviation changes to σ\sigma Ohms. The machine that makes the Type B resistors is not adjusted. The probability that the resistance of a randomly selected resistor from these machines is now less than 115 Ohms is 0.18.

Find the value of σ\sigma.

[5]

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What does Bayes theorem cover in IB Maths AA?

Binomial Distribution: Models the number of successes in a fixed number of independent trials with constant success probability (n, p). Key stats: Mean = np, Variance = np(1-p). Use Binomial PDF for exact probabilities and CDF for cumulative ones. Normal Distribution: A continuous distribution, denoted X backsim N(μ, σ^2). Characterized by its bell curve. Standardize using Z = (X - μ) / σ to get Z backsim N(0, 1). Use Normal CDF to find probabilities (areas) and Inverse Normal to find values given probabilities. Empirical rule: ~68% within μ ± σ, ~95% within μ ± 2σ, ~99.7% within μ ± 3σ.

Is Bayes theorem SL or HL?

Bayes theorem is HL only. SL students are not examined on it.

How do I revise Bayes theorem for IB Maths AA?

Start from the core idea: binomial Distribution: Models the number of successes in a fixed number of independent trials with constant success probability (n, p). Key stats: Mean = np, Variance = np(1-p). Use Binomial PDF for exact probabilities and CDF for cumulative ones. In the exam: three events is the ceiling, so a question with a four-branch partition is out of syllabus. The medical-testing context (false positives) is the archetype. Then practise exam-style questions, easiest first, writing out every step of your working before you check it.

How does FourtyFive help me practise Bayes theorem?

FourtyFive has 3 Bayes theorem questions. Every answer you write is marked mark by mark, IB-style, and you see where each mark was won or lost. Every part has a hint, the AI tutor helps you through the step you are stuck on, and your Study Profile picks what to practise next.

Is FourtyFive free for Bayes theorem practice?

Yes. A free account gives you 50 marked answers a month, and you do not need a card to sign up.

Can I handwrite Bayes theorem answers on an iPad?

Yes. In the FourtyFive iPad app you write your working by hand with Apple Pencil, the way you would on paper, and it is marked the same way.

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