Relative and Expected Frequency - Worksheets, Questions and Revision

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GCSE · Statistics

S23 Relative and Expected Frequency

EDEXCEL 1ST0 · Calculator allowed · about 80 minutes
Total Marks
Name: _______________________________    Date: ____ / ____ / ______
Answer ALL questions. Show all your working.
1
Four key terms are used when studying probability experiments: trial, relative frequency, theoretical probability, expected frequency.

For each description below, write down the term being described.
(a)One single go at an experiment, such as one throw of a dice or one spin of a spinner.(1)
(b)The value found by working out (number of times an outcome occurs) divided by (total number of trials).(1)
(c)A probability worked out using known facts about a fair situation, without needing to carry out an experiment, e.g. 1/6 for one particular score on a fair dice.(1)
(d)The value found by working out (probability of an outcome) multiplied by (number of trials).(1)
(Total for Question 1 is 4 marks)
2
An ordinary fair six-sided dice is rolled 40 times. The table shows the number of times each score occurs.
Score123456
Frequency576895
(a)Write down the total number of trials shown in the table.(1)
(b)Calculate the relative frequency of rolling a 5. Give your answer as a decimal.(2)
(c)Write down which score has the highest relative frequency.(1)
(Total for Question 2 is 4 marks)
3
The bar chart shows the results when a four-colour spinner (red, blue, green and yellow) was spun 200 times.
0 10 20 30 40 50 60 Red Blue Green Yellow Purple Colour Number of spins
(a)Calculate the relative frequency of the spinner landing on yellow. Give your answer as a decimal.(2)
(b)Write down which colour was the least likely outcome in these 200 spins.(1)
(c)Using the relative frequency from part (a), estimate how many times the spinner would land on yellow in 500 spins.(2)
(Total for Question 3 is 5 marks)
4
A newsagent recorded the number of magazines sold in one hour. Of the 60 magazines sold, 15 were sports magazines.
(a)Calculate the relative frequency that a magazine sold was a sports magazine. Give your answer as a fraction in its simplest form.(2)
(b)The newsagent expects to sell 300 magazines next week. Use the relative frequency from part (a) to estimate how many of these will be sports magazines.(2)
(c)State one assumption needed for the estimate in part (b) to be reliable.(1)
(Total for Question 4 is 5 marks)
5
A biased four-sided spinner has a probability of 0.3 of landing on red. The spinner is spun 250 times.
(a)Work out the expected number of times the spinner lands on red.(2)
(b)Write down the expected number of times the spinner does not land on red.(1)
(Total for Question 5 is 3 marks)
6
For each statement about relative frequency and expected frequency, write down whether it is True or False.
(i)The relative frequency of an outcome can be greater than 1.(1)
(ii)As the number of trials in an experiment increases, the relative frequency generally gets closer to the true probability.(1)
(iii)The expected frequency of an outcome always equals the number of times it actually occurs.(1)
(iv)Relative frequency can be used to estimate a probability when the theoretical probability is not known.(1)
(Total for Question 6 is 4 marks)
7
The spinner shown is divided into 3 unequal sections, A, B and C, based on the angles marked at the centre. The spinner is spun 300 times.
A B C
(a)Write down the probability that the spinner lands on C. Give your answer as a fraction in its simplest form.(2)
(b)Work out the expected number of times the spinner lands on B in the 300 spins.(2)
(c)Work out the expected number of times the spinner lands on A or C combined in the 300 spins.(2)
(Total for Question 7 is 6 marks)
8
A coin is tossed repeatedly. The table shows the cumulative number of heads recorded after different numbers of tosses.
Number of tosses1050100500
Number of heads72352248

Give each relative frequency correct to 3 decimal places.
(a)Calculate the relative frequency of heads after 10 tosses.(1)
(b)Calculate the relative frequency of heads after 50 tosses.(1)
(c)Calculate the relative frequency of heads after 100 tosses.(1)
(d)Calculate the relative frequency of heads after 500 tosses.(1)
(e)Comment on what these relative frequencies suggest about the probability that this coin lands on heads.(2)
(Total for Question 8 is 6 marks)
9
A games manufacturer wants to check whether a new four-sided spinner, used in a board game, is fair. The spinner is labelled 1, 2, 3 and 4, and should give an equal chance of landing on each number.
(a)State the stage of the statistical enquiry cycle at which the manufacturer decides how many times to spin the spinner before testing it.(1)
(b)State the stage of the statistical enquiry cycle at which each spin's result is written down in a tally chart.(1)
(c)The spinner is spun 400 times. It lands on 4 exactly 130 times. State the stage of the statistical enquiry cycle at which the manufacturer would calculate the relative frequency of landing on 4, and calculate this relative frequency. Give your answer to 3 decimal places.(3)
(d)The theoretical probability of landing on 4 for a fair spinner is 1/4 = 0.25. State the stage of the statistical enquiry cycle at which the manufacturer would draw a conclusion, and comment on whether the spinner appears to be fair.(2)
(Total for Question 9 is 7 marks)
10
A survey asked 120 students at a school whether they prefer Maths or Art. The two-way table shows the results.
MathsArtTotal
Boys283260
Girls421860
Total7050120
(a)Calculate the relative frequency that a student chosen at random from this survey prefers Art.(2)
(b)Calculate the relative frequency that a girl in this survey prefers Maths.(2)
(c)The school has 900 students in total, and the proportions in this survey are believed to be representative of the whole school. Use the relative frequency from part (a) to estimate how many of the 900 students prefer Art.(2)
(Total for Question 10 is 6 marks)
11
A factory tests a random sample of 80 light bulbs taken from a production run of 5000 bulbs. In the sample, 6 of the bulbs are found to be faulty.
(a)Calculate the relative frequency of a bulb in the sample being faulty. Give your answer as a decimal.(2)
(b)Use the relative frequency from part (a) to estimate the number of faulty bulbs in the full production run of 5000 bulbs.(2)
(c)The factory will recall the whole production run if the estimated number of faulty bulbs is more than 300. Using your answer to part (b), state whether the factory should recall the production run, giving a reason.(1)
(Total for Question 11 is 5 marks)
12
At a school fete, a stallholder runs a game. Players pay £2 to play. Each player picks one ball at random from a bag of 50 balls, of which 4 are winning balls. A player who picks a winning ball receives a £10 prize. 250 people play the game during the day.
(a)Work out the probability that a single player picks a winning ball. Give your answer as a fraction in its simplest form.(2)
(b)Work out the expected number of winning picks out of the 250 players.(2)
(c)Work out the total amount taken in entry fees from the 250 players.(1)
(d)Using your answers to parts (b) and (c), work out the stallholder's expected profit for the day.(2)
(Total for Question 12 is 7 marks)
13
Two students each test the same coin to estimate the probability that it lands on heads. Student A tosses the coin 20 times and gets 14 heads. Student B tosses the coin 200 times and gets 118 heads.
(a)Calculate Student A's relative frequency of heads.(1)
(b)Calculate Student B's relative frequency of heads.(1)
(c)Explain which student's result gives the more reliable estimate of the probability that this coin lands on heads.(2)
(d)Using the more reliable estimate from part (c), work out the expected number of heads in 350 tosses of this coin.(2)
(Total for Question 13 is 6 marks)
14
An insurance company reviewed 400 car insurance policies from last year, split by the driver's age group and whether a claim was made. The two-way table shows the results.
Claim madeNo claimTotal
Under 25184260
25 and over34306340
Total52348400
(a)Calculate the relative frequency that a policy in the Under 25 group had a claim made.(2)
(b)The company sells 500 new policies to under-25 drivers next year. Using the relative frequency from part (a), work out the expected number of these new policies that will have a claim made.(2)
(c)Each claim made by an under-25 driver costs the company an average of £1200. Work out the expected total cost of claims from the 500 new under-25 policies.(2)
(d)Suggest one reason, based on the data in the table, why the insurance company might charge under-25 drivers a higher premium than older drivers.(1)
(Total for Question 14 is 7 marks)
15
A biased dice has a probability of 0.15 of showing a six. The dice is rolled n times, and the expected number of sixes is 27.
(a)Form an equation using n, and solve it to find the number of times the dice was rolled.(2)
(b)The dice is actually rolled 180 times and a six occurs 34 times. Determine, with a reason, whether this result supports the claim that the probability of a six is 0.15 for this dice.(3)
(Total for Question 15 is 5 marks)
Mark scheme · S23 Relative and Expected Frequency

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Question 15