L3 Probability Concepts
Question Bank
An events venue near Nelson holds functions (weddings, birthdays and work parties) either on its outdoor lawn or in its indoor hall.
- 40% of functions are booked for the outdoor lawn.
- It rains on the day of 30% of functions.
Assume that whether a function is booked for the lawn is independent of whether it rains on the day.
(a) Calculate the probability that a randomly chosen function is booked for the lawn and it rains on the day.
(b) The venue holds 150 functions a year. About how many of them would you expect to be booked for the lawn and have rain on the day?
A tramping club asked its 120 members which of two multi-day tracks they have walked: the Abel Tasman Coast Track and the Routeburn Track. The Venn diagram shows the results.
Explain whether the events ‘member has walked the Abel Tasman Coast Track’ and ‘member has walked the Routeburn Track’ are mutually exclusive.
A hair salon recorded, for 200 customers, whether they booked online or walked in, and whether they were seen within 15 minutes of arriving.
| Seen within 15 min | |||
|---|---|---|---|
| Booking | Yes | No | Total |
| Online | 48 | 32 | 80 |
| Walk-in | 52 | 68 | 120 |
| Total | 100 | 100 | 200 |
The salon wants to know whether being seen within 15 minutes is independent of booking online. As a first step:
(a) Calculate the probability that a randomly chosen customer was seen within 15 minutes.
(b) A customer is randomly selected from those who booked online. Calculate the probability that they were seen within 15 minutes.
A beef farm recorded the 160 calves born on it this spring: the breed of each calf (Angus, Hereford or Simmental), and whether it was kept on the farm or sold.
- Of the 90 calves that were kept, 30 were Angus and 40 were Hereford.
- Of the 70 calves that were sold, 44 were Angus and 16 were Hereford.
- The rest of the calves were Simmental.
(a) Explain whether the events ‘calf is an Angus’ and ‘calf is a Hereford’ are mutually exclusive.
(b) Calculate the probability that a randomly chosen calf from this spring was an Angus or a Hereford.
In March, 60 000 people used a city’s public transport card. Of these card users:
- 8400 took at least one ferry trip
- 31 500 took at least one bus trip
- 5600 took both a ferry trip and a bus trip.
(a) A card user is chosen at random from those who took a ferry trip. Calculate the probability that they also took a bus trip.
(b) Explain whether the events ‘card user took a ferry trip’ and ‘card user took a bus trip’ are mutually exclusive. Support your answer with statistical reasoning.
At a pet shop, 45% of customers buy pet food, and 30% of customers belong to the shop’s loyalty club.
The manager predicts that 38.5% of customers will be people who are not in the loyalty club and do not buy pet food.
Show how the manager made this prediction, including stating any assumption(s) that were made.
Over several seasons, half of a basketball team’s games have been at home. The team has won 75% of its home games and 50% of its away games.
Calculate the probability that the team wins a randomly chosen game, and use it to explain whether the events ‘game is played at home’ and ‘team wins the game’ are independent.
A car yard has 30 cars for sale. Each car is either a hybrid or a petrol car, and either an SUV or a hatchback.
- 18 of the cars are hybrids.
- 14 of the cars are SUVs.
- 2 of the hybrids are SUVs.
Explain whether the events ‘car is a petrol car’ and ‘car is a hatchback’ are mutually exclusive. Give at least one numerical calculation.
At a guitar school, 60% of students learn in a group class and the rest have private lessons. 30% of the group-class students and 50% of the private-lesson students can play a whole song by the end of their first month.
A tutor says that the probability that a randomly chosen student is in a group class and can play a whole song, added to the probability that a randomly chosen student has private lessons and cannot play a whole song, must equal 1.
Explain, using statistical reasoning, whether the tutor is correct.
A plumbing company has found that:
- 10% of its jobs take longer than the time quoted
- 30% of its jobs are in houses built before 1950
- 60% of the jobs that took longer than the time quoted were in houses built before 1950.
Comment on whether the events ‘job takes longer than the time quoted’ and ‘house was built before 1950’ are independent of each other. Use statistical reasoning to support your answer.
AND interpret your answer in terms of the relationship between the events ‘job takes longer than the time quoted’ and ‘house was built before 1950’.
A supermarket looked at 200 of its online grocery orders and recorded which of three items each order included: eggs, flour and butter. The Venn diagram shows the results.
(a) Calculate the probability that a randomly chosen order included both eggs and flour.
(b) Interpret the relationship between the events ‘order includes eggs’ and ‘order includes flour’. As part of your answer, explain whether these two events are independent. Use calculations and statistical reasoning to support your answer.
A council surveyed 250 households in its district:
- 100 households have a compost bin
- 150 households use the council’s food-scraps collection
- 30 households have a compost bin and use the food-scraps collection.
(a) What proportion of these households have a compost bin but do not use the food-scraps collection?
(b) The council claims that having a compost bin has no effect on whether a household uses the food-scraps collection. Investigate the validity of this claim. Use statistical reasoning to support your answer.