L2 Probability

A university department conducted a survey on 500 first-year students to study the relationship between where they studied (library or home) and their exam result (Pass or Fail).

(i) What is the probability that a randomly chosen student passed the exam?

(ii) What is the probability that a randomly chosen participant in this survey who studied at the library passed?

(iii) A concerned lecturer notices that most students in the library spend more time passing notes to their friends than studying. He claims that students studying at the library are twice as likely to fail than students who study at home. Evaluate this claim, supporting your answer with calculations.

A recent survey was conducted in a major city to gauge public interest in electric scooters. The survey randomly sampled 850 adults, asking if they owned an electric scooter and their gender.

(i) What proportion of the people surveyed were female and did not own an electric scooter?

(ii) If a person selected at random owns an electric scooter, what is the probability that they are male?

(iii) The city council suggests that males are more than twice as likely as females to own an electric scooter. Use a calculation to determine if this claim is statistically supported by the data. Interpret your result in context.

A weather service tracked 1,500 forecasts over a year, categorizing them by the forecast range (Short-Range: next 48 hours, or Long-Range: next 7 days) and the final outcome.

Table 1: Forecast Accuracy vs. Range

(i) A new weather app advertises that a Long-Range forecast is more than 3 as likely to be incorrect as a Short-Range forecast. Is this claim supported by the data? Show your calculation and interpretation.

(ii) Another meteorologist asserts that the chance of a forecast being Correct is about 60% higher for a Short-Range forecast than for a Long-Range forecast. Evaluate this assertion using an appropriate calculation.

A vet clinic followed 600 dogs until they were eight years old. It recorded whether each dog was walked daily and whether it developed joint problems (“Sore joints” in the table).

• 60% of the dogs were walked daily.
• Half of the dogs that developed joint problems were walked daily.

(i) Use the information above to complete the table.

(ii) A pet magazine claims that a dog that is not walked daily is 80% more likely to develop joint problems than a dog that is walked daily.

Use the data to evaluate this claim. Use relative risk, and interpret it in context.

A tramping club recorded the footwear of 320 members on a three-day tramp, and whether they got blisters.

The club secretary says, “Boots are more likely to give you blisters. 48 people in boots got blisters, but only 28 people in trail shoes did.”

(i) Explain why the secretary’s reasoning is flawed.

(ii) Use relative risk to show whether boots or trail shoes were linked with a higher risk of blisters on this tramp. Interpret your answer in context.

A health researcher collected data on the sleep habits of 650 adults. The table shows the proportions of participants broken down by average sleep duration and exercise frequency.

Table 1: Sleep Duration and Exercise Frequency

(i) What is the probability that a randomly chosen participant exercises occasionally, given that they sleep less than 8 hours?

(ii) What proportion of participants sleep 8 hours or more and exercise regularly?

(iii) A fitness influencer claims that getting regular excercise increases the chance you'll sleep longer by a lot. Discuss this claim, along with comparing any relevant probabilities in an appropriate way.

A university trial studied the pass rates of a large introductory course, comparing the method of instruction with the final result. A sample of 850 students gave the following results:

Table 1: Course Method vs. Result

(i) A university administrator claims that students who study In-Person are less than half as likely to fail the course as students who study Online. Use a calculation to test this claim. Interpret your result in context.

(ii) Another study suggests that the proportion of students who Fail is $1.5$ times greater for those who study Online compared to those who study In-Person. Determine if the data in the table supports this claim.

A school trialled a reading app with its 480 Year 9 students. At the end of the year, every student sat a reading check. Three-fifths of the students used the app.

(i) Complete the table.

(ii) The school newsletter says, “Using the app halves a student’s chance of failing the reading check.”

Use relative risk to evaluate this claim, and interpret your answer in context.