Probability experiments & frequency

A student has a three-sided spinner with sides labelled A, B and C. They spin the spinner a number of times and record the results.

The student works out the correct relative frequencies for the spinner landing on sides A and B. The relative frequencies are shown on this scale.

Write down the relative frequency that the spinner landed on side A.

The student then works out the relative frequency that the spinner landed on side C is 0.4.

Explain, using a calculation, why the relative frequencies show that the student has made a mistake.

The student says

The probability that the spinner will land on side B on my next spin is 0.1.

What assumption is the student making?

A spinner has five sides numbered 1 to 5. If the spinner is fair, the probability that it lands on the number 1 is 0.2.

A student spins the spinner 300 times.

Assuming the spinner is fair, use the information to work out how many times the spinner is expected to land on the number 1.

The spinner actually landed on the number 1 on 58 of the 300 spins.

Decide whether or not the result suggests this spinner is likely to be a fair spinner? Give a reason for your answer.

In a particular town last year:

• it rained on 17 of the 30 days in November

• it rained on 18 of the 31 days in December.

Which month, November or December, had the highest proportion of rainy days? Show how you decide.

Sam says, I think the probability it will rain on December 25th next year is What assumption has Sam made?

A bag only contains green, red, blue and yellow discs.

Orla carries out an experiment. She picks one disc at a time from the bag, records its colour and then returns the disc to the bag. When she has finished the experiment, Orla works out the relative frequency of each colour. Some of her results are shown in the table.

The relative frequency of the yellow discs was three times the relative frequency of the blue discs.

In total, there are 2000 discs in the bag.

Use this information to find an estimate for the total number of green and yellow discs that are in the bag. You must show your working.

Explain why your estimate may not be reliable.

Jack travels to work each day by train.

He records whether

• the train is on time or late

• there are seats available or no seats available.

Jack’s results are shown on this partly completed frequency tree.

Find the relative frequency of there being no seats available on Jack’s train journey.

Jack says

If the train is late, travellers are less likely to find seats available than if the train was on time.

Does Jack’s data suggest he is correct? Show how you decide.

Gabi records the number of times a biased six-sided dice lands on each of its numbers.

Complete the table to show the relative frequencies.

Use Gabi’s results to estimate the probability that the spinner lands on 5 or 6.

A bag contains 150 counters. The counters are either red or yellow.

Riley picks a counter from the bag, records its colour, and replaces it. They do this nine times.

Here are Riley’s results.

Use Riley’s results to work out how many red counters are likely to be in the bag.

(a)........ red counters

Ling uses the same bag of counters and picks the counters in the same way.

Here are Ling’s results.

Use Ling’s results to estimate the probability of choosing a red counter from the bag. Give your answer as a fraction in its simplest form.

Explain why Ling’s results are likely to give a better estimate of the probability of choosing a red counter from the bag than Riley’s results.

One morning Harper records the first 50 vehicles to pass the school gate. Harper’s results are shown in this table.

Use Harper’s results to estimate the probability that the next vehicle will not be a motorbike.

One afternoon Reece records some vehicles that pass the school gate. of the vehicles they record are cars. For Reece’s results, write down the ratio of cars to not cars. Give your answer in its simplest form.

Kai has a bag of marbles that are red or blue or green or yellow.

Kai takes a marble at random, records the colour and returns the marble to the bag. Kai does this 800 times.

The table shows some of the results.

Complete the table to show the number of times a yellow marble is taken.

There are 40 marbles in the bag.

Work out how many blue marbles are likely to be in the bag.

Is your answer to part (b)(i) likely to be the actual number of blue marbles in the bag? Give a reason for your answer.

A phone manufacturer records the faults that are reported. Last week, in a batch of 96 phones, 6 were reported as faulty.

Write down the relative frequency of faulty phones in this batch.

In 2020, the manufacturer sold a total of 12 321 phones.

Work out how many of these phones the manufacturer should expect to be reported as faulty.

Morgan is playing a computer game. They can score 0, 1, 2 or 3 points on each turn. They record their scores for 100 turns. The table shows the relative frequencies of their scores.

Complete the table.

Morgan says

I scored more than 160 points in total in my 100 turns.

Is Morgan correct? Show how you decide.

Morgan is playing a computer game. They can score 0, 1, 2 or 3 points on each turn. They record their scores for 100 turns. The table shows the relative frequencies of their scores.

Complete the table.

Morgan says

I scored more than 160 points in total in my 100 turns.

Is Morgan correct? Show how you decide.

Ling throws a six-sided dice 300 times. The table shows the frequencies of their results.

Complete the table to show the relative frequencies.

Ling thinks that the dice may be biased.

Explain why evidence from the table could support their opinion.

Explain why the dice may, in fact, not be biased.

Over a long period of time, it is found that the probability of a train from Bewford to London being late is 0.2.

One morning there are two trains from Bewford to London.

Use the information to complete the tree diagram.

Work out the probability that both trains are not late.

Give a reason why the probabilities used in the tree diagram for the second train may not be reliable.

Morgan takes a train from London to Bewford and then another train to Agon. The tree diagram shows the probabilities of Morgan’s trains being late or not late.

Morgan will not catch the train to Agon if the train to Bewford is late and the train to Agon is not late.

Work out the probability that Morgan will catch the train to Agon.

Ling throws a six-sided dice 300 times. The table shows the frequencies of their results.

Complete the table to show the relative frequencies.

Ling thinks that the dice may be biased.

Explain why evidence from the table could support their opinion.

Explain why the dice may, in fact, not be biased.

One morning Kai records the colour of the cars passing his house. He then works out the relative frequency of each colour. Some of his results are shown in this table.

The following morning, Kai is going to record the colour of the first 200 cars to pass his house.

Work out an estimate for the total number of cars, coloured silver or red, that he should expect to see.

Plaza United are playing a football match away from home.

379 supporters are going to the match by coach. Each coach seats 45 people.

What is the smallest number of coaches that will be needed?

In their last 50 matches, Plaza United have drawn 10 matches, lost 5 and won the rest.

Sam claims

The probability that Plaza United will win this match is 0.7.

Show calculations to support Sam’s claim.

Give one reason why Sam’s claim may not be reliable.

Jenny has a five-sided biased spinner. The sectors are coloured red, blue, green, yellow and white. She spins the spinner 100 times.

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

Jenny uses her data to estimate the probability of the spinner landing on each colour.

Write down Jenny’s estimate for the probability of landing on red.

Jenny then writes in some of the colours on this probability scale.

Write the correct colour in the box.

Explain why Jenny’s estimate for the probability of landing on yellow cannot be the actual probability.

A bag contains 100 pencils that are either red or green.

Describe a method you could use to estimate the number of red pencils in the bag without looking into the bag or having more than one of the pencils out of the bag at any one time.

The table shows the relative frequencies of the results for a football team after a number of games.

Complete the table.

The team lost 10 more games than they won.

How many games did the team play altogether?

The table shows the relative frequencies of the results for a football team after a number of games.

Complete the table.

The team lost 10 more games than they won.

How many games did the team play altogether?

A hotel manager asked some people to choose their favourite breakfast fruit juice. They each chose one from Orange, Apple or Grapefruit.

• 20 men chose Orange

• Equal numbers of men chose Apple and Grapefruit.

• 10 women chose Orange

• Twice as many women chose Apple as Grapefruit.

Use this information to complete the frequency tree.

In one week 200 men have breakfast at the hotel.

How many men may be expected to drink Orange?

Give one reason why the number of men who drink Orange in this week may be different to your answer to part (b)(i).

Jeat is growing carrots from seed in his garden. He plants 28 carrot seeds but only 12 grow.

Jeat says The probability of one of my carrot seeds growing is

Use Jeat’s result to show that he is correct.

A farmer uses this probability to calculate how many carrot seeds he should plant to grow 10 000 carrots.

How many seeds should he plant?

Explain why it may not be sensible for the farmer to use Jeat’s experimental probability to calculate the number of seeds he should plant.