Questions: Law of total probability and Bayes’ theorem
Before attempting these questions it is highly recommended that you read Guide: Law of total probability and Bayes’ theorem.
Q1
Use the law of total probability to answer the following.
1.1.
In a hospital:
- \(40\%\) of patients are treated in Ward A,
- \(60\%\) in Ward B,
- the probability of recovery within 3 days is \(80\%\) in Ward A,
- the probability of recovery within 3 days is \(60\%\) in Ward B.
Let \(R\) be the event that a patient recovers in 3 days. What is \(\mathbb{P}(R)\)?
1.2.
A school has three types of lunches:
- \(50\%\) of students choose vegetarian
- \(30\%\) choose chicken
- \(20\%\) choose fish
The probability that a student finishes their lunch is:
- \(90\%\) for vegetarian
- \(70\%\) for chicken
- \(80\%\) for fish
What is the probability that a randomly chosen student finishes their lunch?
1.3.
The magnificent Mersenne Macarons are manufactured in three Cantor’s Confectionery factories:
- \(20\%\) from Factory 1 (with a defect rate \(5\%\))
- \(30\%\) from Factory 2 (with a defect rate \(2\%\))
- \(50\%\) from Factory 3 (with a defect rate \(1\%\))
What is the probability that a randomly chosen Mersenne Macaron is defective?
1.4.
A student can study in three locations:
- At home (\(50\%\) of the time)
- In the library (\(30\%\))
- In a café (\(20\%\))
The probability they complete their homework is:
- \(70\%\) at home
- \(90\%\) in the library
- \(60\%\) in the café
What is the probability that a randomly selected student completes their homework?
Q2
Use Bayes’ theorem to answer the following.
2.1.
Statistics for a test for a disease is:
- \(95\%\) accurate for infected individuals (true positive)
- \(90\%\) accurate for uninfected individuals (true negative)
- \(2\%\) of the population has the disease
Let \(D\) be the event that a person has the disease and \(T\) the event they test positive. What is \(\mathbb{P}(D \mid T)\)?
2.2.
In St Andrews, Scotland:
- \(60\%\) of days are dry
- \(40\%\) are rainy
A forecast predicts rain:
- \(80\%\) of the time on rainy days
- \(10\%\) of the time on dry days
If the forecast predicts rain in St Andrews, what is the probability that it will actually rain?
2.3.
In a Cantor’s Confectionery factory:
- \(70\%\) of Bayes Biscuits are made by Machine A
- \(30\%\) by Machine B
The probability of a broken Bayes Biscuit is:
- \(2\%\) from Machine A
- \(5\%\) from Machine B
If a biscuit is broken, what is the probability it came from Machine B?
2.4.
A brand new bag of Gauss Gummies contains:
- \(40\%\) red sweets
- \(60\%\) blue sweets
A red sweet has a \(30\%\) chance of having a wrapper and a blue sweet has a \(70\%\) chance of having a wrapper. If a sweet is picked at random and has a wrapper, what is the probability it is red?
After attempting the questions above, please click this link to find the answers.
Version history and licensing
v1.0: initial version created 05/25 by Sophie Chowgule as part of a University of St Andrews VIP project.