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IB Mathematics AA Use of Bayes’ theorem Study Notes

IB Mathematics AA Use of Bayes’ theorem Study Notes

IB Mathematics AA Use of Bayes’ theorem Study Notes

IB Mathematics AA Use of Bayes’ theorem Notes Offer a clear explanation of Use of Bayes’ theorem, its mean and variance, including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic Use of Bayes’ theorem, its mean and variance.

Bayes’ Theorem

Introduction

Bayes’ Theorem is a fundamental result in probability theory that allows us to update probabilities based on new evidence. It is widely applied in medical testing, spam filtering, artificial intelligence, and risk assessment.

Key Concepts

1. Bayes’ Theorem Formula

  • Bayes’ Theorem states that for two events \( A \) and \( B \):

\( P(A|B) = \frac{P(B|A) P(A)}{P(B)} \)

  • \( P(A|B) \) = Probability of event \( A \) given event \( B \) (posterior probability).
  • \( P(B|A) \) = Probability of event \( B \) given event \( A \) (likelihood).
  • \( P(A) \) = Prior probability of event \( A \) (before observing \( B \)).
  • \( P(B) \) = Total probability of event \( B \), considering all possible causes of \( B \).

2. Bayes’ Theorem for Multiple Events

  • When there are multiple mutually exclusive and exhaustive events \( A_1, A_2, \dots, A_n \), the probability of \( A_k \) given \( B \) is:

\( P(A_k|B) = \frac{P(B|A_k) P(A_k)}{\sum P(B|A_i) P(A_i)} \)

Guidance, Clarification, and Syllabus Links

  • Bayes’ Theorem will be applied to a maximum of three events.
  • Understanding conditional probability is essential.
  • It links to independent events (SL4.6).

Connections and Applications

1. Other Contexts

  • Medical Studies: Used to assess the probability of diseases based on test results.
  • Machine Learning: Naïve Bayes classifiers are used in spam filtering, sentiment analysis, and recommendation systems.
  • Forensic Science: Helps in DNA matching and legal decision-making.
  • Finance: Used to estimate the probability of market events.

Solved Example

Problem: A certain disease affects 1% of a population. A diagnostic test correctly detects the disease 95% of the time when a person has it, but it also gives a false positive 5% of the time when a person does not have the disease. What is the probability that a person who tests positive actually has the disease?

Step 1: Define Events

  • \( D \) = Event that a person has the disease.
  • \( \neg D \) = Event that a person does not have the disease.
  • \( T \) = Event that the test result is positive.

Step 2: Identify Given Probabilities

  • \( P(D) = 0.01 \) (Prior probability of having the disease).
  • \( P(\neg D) = 0.99 \) (Prior probability of not having the disease).
  • \( P(T|D) = 0.95 \) (Sensitivity: Probability of testing positive given that the person has the disease).
  • \( P(T|\neg D) = 0.05 \) (False positive rate: Probability of testing positive given that the person does not have the disease).

Step 3: Compute Total Probability of Testing Positive

Using the Law of Total Probability:

\( P(T) = P(T|D)P(D) + P(T|\neg D)P(\neg D) \)

\( = (0.95 \times 0.01) + (0.05 \times 0.99) \)

\( = 0.0095 + 0.0495 = 0.059 \)

Step 4: Apply Bayes’ Theorem

\( P(D|T) = \frac{P(T|D) P(D)}{P(T)} \)

\( = \frac{0.95 \times 0.01}{0.059} \)

\( = \frac{0.0095}{0.059} \approx 0.161 \)

Step 5: Interpretation

  • Even though the test detects the disease with high accuracy (95% sensitivity), the probability that a person who tests positive actually has the disease is only 16.1%.
  • This shows the importance of understanding conditional probability and not relying solely on a test’s accuracy.
  • Medical tests must be interpreted alongside prior probabilities and additional diagnostic criteria.

Additional Insights

1. Why is the probability so low?

  • Because the disease is rare (only 1% prevalence), the number of false positives outweighs the true positives.

2. How can this be improved?

  • Using a second independent test with different characteristics.
  • Applying Bayesian updating with additional information (e.g., symptoms, risk factors).

IB Mathematics AA SL Use of Bayes’ theorem Exam Style Worked Out Questions

Question

At a nursing college, 80 % of incoming students are female. College records show that 70 % of the incoming females graduate and 90 % of the incoming males graduate. A student who graduates is selected at random. Find the probability that the student is male, giving your answer as a fraction in its lowest terms.

▶️Answer/Explanation

Markscheme

\({\text{P }}M|G = \frac{{{\text{P}}(M \cap G)}}{{{\text{P}}(G)}}\)     (M1)

\( = \frac{{0.2 \times 0.9}}{{0.2 \times 0.9 + 0.8 \times 0.7}}\)     M1A1A1

\( = \frac{{0.18}}{{0.74}}\)

\( = \frac{9}{{37}}\)     A1

[5 marks]

Examiners report

Most candidates answered this question successfully. Some made arithmetic errors.

Question

Jenny goes to school by bus every day. When it is not raining, the probability that the bus is late is \(\frac{3}{{20}}\). When it is raining, the probability that the bus is late is \(\frac{7}{{20}}\). The probability that it rains on a particular day is \(\frac{9}{{20}}\). On one particular day the bus is late. Find the probability that it is not raining on that day.

▶️Answer/Explanation

Markscheme

     (A1)

 

\({\text{P}}(R’ \cap L) = \frac{{11}}{{20}} \times \frac{3}{{20}}\)     A1

\({\text{P}}(L) = \frac{9}{{20}} \times \frac{7}{{20}} + \frac{{11}}{{20}} \times \frac{3}{{20}}\)     A1

\({\text{P}}(R’|L) = \frac{{{\text{P}}(R’ \cap L)}}{{{\text{P}}(L)}}\)     (M1)

\( = \frac{{33}}{{96}}{\text{ }}\left( { = \frac{{11}}{{32}}} \right)\)     A1

[5 marks]

Examiners report

This question was generally well answered with candidates who drew a tree diagram being the most successful.

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