Probability experiments and frequency

Chiara spins a fair 8‑sided spinner.

Circle the word from the list that best describes the likelihood that the spinner lands on a 5

certain evens likely impossible unlikely

On the probability scale, mark with a cross (×) the probability that the spinner lands on a 4

On the probability scale, mark with a cross (×) the probability that the spinner lands on an even number.

Write down the probability that the spinner lands on a 3

Chiara spins the spinner a total of 80 times.

Work out the expected number of times the spinner will land on a 3

Below are 10 cards from a game.

Complete this two‑way table.

One of the 10 cards is picked at random.

Write down the probability that the card picked is a star.

Claire has a spinner.

She spins the spinner 10 times and records the colour the spinner lands on. It lands on red 3 times.

Using Claire’s results, estimate the probability that the next time she spins the spinner it will land on red.

Dhiyan spins the same spinner 50 times and records the colour the spinner lands on each time.

Here are Dhiyan’s results.

Using Dhiyan’s results, estimate the probability that the next time he spins the spinner it will land on red.

Whose data will give a more reliable estimate for the probability, Claire’s or Dhiyan’s? Give a reason for your answer.

Considering Dhiyan’s results, is the spinner fair? Give a reason for your answer.

Elijah has a coin.

Elijah says, “I predict that the probability of flipping a head on this coin is 50%”

Name the type of probability that Elijah has given.

Peter flips Elijah’s coin 300 times. Here are his results.

Peter says, “The probability of flipping a head on this coin is 46%”

Comment on the differences between these two probabilities.

Jenny is investigating how many days per week people use a gym.

She asks the 40 people in her fitness group how often they use the gym each week. Jenny draws this bar chart for her data.

One of these people is chosen at random.

Find the probability that this person uses the gym exactly 2 days per week.

What is the modal number of days to use the gym each week?

Jenny thinks that there are a lot of people in her fitness group who are exercising less than 2 days per week as there is a total of 10 people who used the gym on 0 days or 1 day per week.

Explain why Jenny might not be correct.

Tachi collects data on the heights, in metres, of a sample of Egyptian pyramids.

Here is her data.

136.4 101.1 104 62.6 138.8 65.5 93.5 (Source: www.rankred.com)

Tachi picks one of these pyramids at random.

Find the probability that this pyramid will have a height of more than 100 m.

The mean height of a sample of Mexican pyramids is 53.5 m.

Tachi says, “On average these Egyptian pyramids are twice as high as the Mexican pyramids.”

Is Tachi correct? You must show working to support your answer.

The range of heights for the Mexican pyramids is 45 m. The lowest height of the Mexican pyramids is 30 m.

Work out the greatest height of the Mexican pyramids.

Jonathan has a fair 6-faced dice which has the numbers 1, 2, 3, 4, 5 and 6 on its faces.

Jonathan rolls his dice once.

On the probability scale below, mark with a cross (×) the probability that the dice will land on an odd number.

0 50% 100%

Kasia has a fair 8-faced dice which has the numbers 1, 2, 3, 4, 5, 6, 7 and 8 on its faces.

Circle the word from the list below that best describes the likelihood that the dice lands on a 9

impossible unlikely evens likely certain

Kasia rolls her dice 80 times.

Work out the number of times you would expect her dice to land on a 5

Jonathan is going to roll his dice once. Kasia is going to roll her dice once.

Is Jonathan more likely to roll a 6 than Kasia? You should justify your answer.

Norbert asked each of the students in his class to name their favourite fruit from Apple, Banana, Orange or Pear.

The results are shown below.

Banana Orange Apple Banana Pear Apple

Apple Banana Orange Pear Apple Banana Apple

Apple Apple Orange Apple Pear Banana Banana

Fill in the tally chart for this information and complete the frequency column.

How many students are in the class?

One of the students is chosen at random.

Find the probability that this student’s favourite fruit is Orange.

Compare the number of students whose favourite fruit is Apple to the number of students whose favourite fruit is Pear.

Norbert decides to find the favourite fruit that is the mode.

Explain why the mode is an appropriate average for Norbert to find for this type of data.

Give one advantage of the tally chart over the raw data.

Norbert wants to draw a diagram to represent his results.

Keshav has a spinner with equal sections numbered 1, 2, 3 and 4

To investigate whether or not the spinner is biased towards the number 1 he spins the spinner 40 times.

Explain what is meant by ‘biased towards the number 1’

Here is information about Keshav’s results.

Keshav says the results show that the spinner is biased.

Discuss whether or not the information in the table supports what Keshav says.

What could Keshav do to help improve the accuracy of his investigation?

Linzi is the owner of a coffee shop and makes afternoon teas for customers. The customers have an option of egg or ham sandwiches and an option of plain or fruit scones. The incomplete two‑way table shows information about the number of afternoon teas she makes one Saturday.

Complete the two‑way table.

One of the customers is chosen at random.

Write down the probability that this customer

ordered a plain scone,

ordered an egg sandwich and a fruit scone,

did not order a ham sandwich.

Linzi needs to place an order for scones for the next Saturday.

Use the information in the table to help her decide if she should order more fruit scones than plain scones. Give a reason for your answer.

Rahul has a 5-sided spinner.

The spinner is a fair spinner.

Explain what the word fair means in this description of the spinner.

Rahul is going to spin his spinner once.

Underline the word from the list below that best describes the likelihood that the spinner will land on 3

impossible unlikely evens likely certain

Underline the word from the list below that best describes the likelihood that the spinner will land on 4

impossible unlikely evens likely certain

Chloe has a 6-faced dice.

She rolls the dice 60 times and records the number on which the dice lands each time.

Here is information about Chloe’s results.

Explain what the information in the table tells you about whether the dice is fair.

Complete the labelling on the frequency axis.

20 employees work in the warehouse in the Rotherham branch.

Complete the key.

In the York branch, 25 employees work in the office and 10 employees work in the warehouse.

Complete the composite bar chart for the York branch.

Find the total number of employees who work in the offices of Northern Stationery Supplies.

Northern Stationery Supplies has a total of 125 employees.

One employee from Northern Stationery Supplies is chosen at random.

Find the probability that this employee works in the warehouse in the Manchester branch.

The following four sentences are about different events.

A The likelihood of an event is evens.

B An event is certain to happen.

C When a fair dice numbered 1 to 6 is rolled, it lands on a number greater than 2

D The next birthday of a person picked at random is on a Monday.

The number line below has arrows marking five probabilities.

Write the letter for each event on the arrow that is in the best position to show the probability of each of the four events.

Eddie has a biased coin.

He wants to find an estimate for the probability that when he throws the coin it will land Heads.

Explain briefly how he could do this.

On her drive to work Hinata passes through two sets of traffic lights. The traffic lights work independently.

On any day that Hinata drives to work the probability that she has to stop at the first set of traffic lights is 0.6 the probability that she has to stop at the second set of traffic lights is 0.5

Complete the probability tree diagram for this information.

Find the probability that Hinata does not have to stop at the first set of traffic lights and does not have to stop at the second set of traffic lights on her way to work.

Each week, Hinata drives to work on 5 days.

She says that she can expect to drive to work without having to stop at either set of traffic lights on exactly one day of the week.

Is Hinata correct?

You must show how you get your answer.

Bryce and Mary have a spinner that can land on yellow or on blue or on red. They want to estimate the probability that when the spinner is spun it will land on red.

Bryce spins the spinner 5 times and the spinner lands on red exactly 1 time.

Bryce used his results to find an estimate for the probability that the next time the spinner is spun it will land on red.

Write down Bryce’s estimate.

Mary spins the spinner 30 times. She records the colour the spinner lands on each time.

The table shows the number of times that the spinner landed on each colour.

Mary used her results to find an estimate for the probability that the next time the spinner is spun it will land on red.

Write down Mary’s estimate.

Whose estimate is more reliable, Bryce’s or Mary’s? Give a reason for your answer.

Chantal claims that the spinner has the same probability of landing on each colour when spun and so is a fair spinner.

Compare Mary’s results with the expected results if Chantal’s claim is true.

The multiple bar charts show information about the percentages of the population in each of Japan and of Kenya that are in each of five different age groups.

(Source: https://www.cia.gov/library/publications/the-world-factbook)

One person is selected at random from the population of Japan.

Work out the probability that this person is not in the age group 15–24

Compare the percentage of the population of Japan aged 0–14 with the percentage of the population of Kenya aged 0–14 Justify your answer using values from the multiple bar charts.

There are two statements below.

Decide whether each statement is true for Japan only or for Kenya only or for both countries or for neither country. Justify each answer using information from the multiple bar charts.

“The age group 55–64 has more than 10% of the population.”

The statement is true for_........

“There is only one age group with a smaller population than the age group 55–64”

The statement is true for_........

Craig thinks that the information in the multiple bar charts could be represented using a population pyramid.

Explain whether or not this would be a suitable diagram for Craig to use.

An activity centre sells Adult tickets and Child tickets. A ticket can be a Day pass or a Weekend pass.

The incomplete two‑way table gives some information about the number of each type of ticket sold last Saturday.

Complete the two‑way table.

One of the people at the activity centre last Saturday with an Adult ticket is selected at random.

Write down the probability that their ticket is a Day pass.

Compare the number of Day passes sold and the number of Weekend passes sold. You should make two comparisons.

There are two types of books, paperback books and hardback books, in a library. The library database records the type of book and when the book was last borrowed.

A book is selected at random from the database.

The librarian draws the probability tree diagram shown below. The tree diagram gives the probability of each type of book. The tree diagram also gives the probability of whether the book was borrowed in the last month.

Write down the probability that a paperback book was borrowed in the last month.

There are 3600 paperback books in the database.

Work out how many of these paperback books were borrowed in the last month.

A book is selected at random from the database.

Find the probability that the book was borrowed in the last month.

Emily and Jess work in the head office of a chain of restaurants.

The menu in all of the restaurants has recently been changed. Emily and Jess have been asked to find out the effect of these changes.

They are going to start by finding out what the kitchen staff think of the new menu.

Emily thinks that they should use a cluster sample of kitchen staff in the restaurants. The restaurants would be the clusters.

Jess thinks that they should use judgement sampling. The head chef of the company would be asked to select suitably experienced kitchen staff for the judgement sample.

Discuss whether these two sampling methods are suitable for Emily and Jess to use.

A website gives diners the opportunity to rate their dining experience at restaurants. Here is one of the questions that diners are asked to answer.

Poor Excellent

1 2 3 4 5 Mark with a cross (X) on the scale, one of the numbers 1 (poor), 2, 3, 4, 5 (excellent) to show how you rate your meal at the restaurant.

Emily recorded the mean ratings, correct to one decimal place, for meals at each of the 28 restaurants in the chain.

The stem and leaf diagram gives information about Emily’s recorded mean ratings.

Emily picks at random one of the restaurants.

Find the probability that the mean rating recorded by Emily for meals at this restaurant is less than 2.5

Jess separates the ratings into those that were made for meals from the previous menu and those that were made for meals from the new menu.

Jess uses her data to draw a percentage composite bar chart in order to compare the ratings for the previous menu and the ratings for the new menu.

The table below gives the percentages for each rating for the new menu.

Use the information in the table to complete the percentage composite bar chart for the new menu.

Use information from the composite percentage bar charts to describe what conclusions can be made about the effects of the changes to the menu.

Some students in Year 9, in Year 10 and in Year 11 at a school were asked to name their favourite meal.

The two-way table gives some information about the favourite meal of each of these students.

Write down the number of Year 9 students whose favourite meal is lunch.

Complete the two-way table.

One of the students is selected at random.

Which meal is least likely to be this student’s favourite meal?

Write down the probability that this student is in Year 11

Find the probability that this student’s favourite meal is either lunch or dinner.

Given that this student is in Year 9, find the probability that this student’s favourite meal is breakfast.

Richard finds information from the internet about the most popular colours for cars in the UK in 2017

Table 1 gives this information.

Explain why the information in Table 1 is secondary data.

Richard drew this pictogram to represent his information about the most popular car colours.

Complete this key for the pictogram.

Jill says that a pictogram is not the most appropriate way to display Richard’s information.

Is Jill correct? Give a reason for your answer.

Richard decides to collect some data about car colours by recording the colour of each of the 60 cars in a car park.

Table 2 gives information about his results.

The modal car colour from Table 2 is blue.

Explain whether or not blue is the expected modal car colour based on the information in Table 1

Explain why, for the information in Table 2, the mode is the most appropriate average to use.

Using the information in Table 1 and Table 2, compare the number of silver cars in the car park with the expected number of silver cars in a car park with 60 cars.

The table gives information about the numbers of students from different types of schools who applied to Cambridge University in 2016

(Source: www.cam.ac.uk)

Richard is going to take a sample of 200 of these students stratified by gender.

Work out how many female students there should be in this sample.

Describe a situation when it would not be appropriate to take a sample stratified by gender.

Richard could have used a different category for his stratified sample.

What is this different category?

A student is to be chosen at random from the 9963 students.

F is the event that the student chosen is female.

I is the event that the student chosen is from an independent school.

M is the event that the student chosen is from a maintained school.

Explain why the event F and the event I are not mutually exclusive.

Find P(I or M).

A company wants to investigate the number of sick days its employees have off work.

The company uses a questionnaire.

Here is one of the questions on the questionnaire.

Assess the appropriateness of the method the company uses.

All the company’s employees completed the question.

615 ticked box A.

102 ticked box B.

Show that an estimate of the number of employees who ticked box A because they answered yes to the question is 137