The RobertSmythSchoolTopic 9
Mathematics Faculty Relative Frequency
Homework onRelative Frequency - Higher
1.Two fair spinners are used for a game. The scores from each spinner are added together.
Tom played the game 500 times and kept a record of how many times he scored a total of 7.He recorded his results in this table.
Number of spins / Total number of 7s / Relative frequencyfirst 50 / 13 / 0.26
first 100 / 23 / 0.23
first 150 / 27
first 200 / 38
first 500 / 105
(a)Complete the table by calculating the relative frequencies for 150 spins, 200 spins and500 spins.
(2)
(b)Which of these relative frequency results gives the best estimate of the probability of scoring a total of 7?Give a reason for your answer.
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(2)
2.Sangita is playing ‘heads or tails’ with her friend.
She spins a fair coin four times and gets four heads.
(a)What is the probability that she gets a tail with her next spin of the coin?
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(1)
(b)If Sangita spins the coin 600 times in succession, approximately how many times should she expect to get a tail?
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(2)
3.A dice is suspected of bias.
Here are the results of 20 throws.
4 / 3 / 1 / 1 / 6 / 2 / 5 / 6 / 5 / 3
(a)Use these results to calculate the relative frequency of each score.
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Score / 1 / 2 / 3 / 4 / 5 / 6Relative frequency
(2)
(b)Use the relative frequency to calculate how many times you would expect to score 3 in 60 throws of this dice.
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Answer ......
(2)
(c)Compare your answer to part (b) with the number of times you would expect to score 3 in 60 throws of a fair dice.
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(1)
4.A fair 5-sided spinner is made from a regular pentagon and labelled as shown.
(a)The spinner is spun 200 times.
How many times would you expect the spinner to land on the letter A?
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(1)
(b)The spinner is spun three times.
Each letter it lands on is recorded.
What is the probability that the word ACE is spelled out?
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(2)
(c)A different 5-sided spinner made from a regular pentagon is tested.
The number of Psis recorded for various numbers of spins.
The results are
Number of spins / Number of letter Ps5 / 1
10 / 3
50 / 17
500 / 152
Is this spinner fair?
Explain your answer fully.
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(2)
5.A sack contains a number of gold and silver discs. An experiment consists of taking a disc from the sack at random, recording its colour and then replacing it. The experiment is repeated 10, 50, 100, 150 and 200 times. The table shows the results.
Number of experiments / 10 / 50 / 100 / 150 / 200Number of gold discs / 3 / 8 / 23 / 30 / 38
(a)Draw a graph to show how the relative frequency of a gold disc changes as the number of experiments increases.
(3)
(b)The sack contains 1 000 discs.Estimate the number of gold discs in the sack.
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Answer ......
(1)
6.Geoff throws a coin 70 times.
He plots the relative frequency of the number of tails after every 10 throws.
(a)How many tails were obtained in 50 throws?
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Answer ...... tails
(2)
(b)Use the diagram to estimate the probability of obtaining a tail.
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Answer ......
(1)
(c)Do you think the coin was biased?
Give a reason for your answer.
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(1)
Mark Scheme
1.(a)0.18, 0.19, 0.21B2
–1 eeoo
(b)0.21B1 ft
ft if prob is between 0 and 1. Accept ‘first 500’ or ‘105 sevens’.
Valid statement eg greatest number of trials
Accept 0.19 and/or 0.21 if reason refers to theoretical probability of 0.2. Invalid reason (eg 0.21 is median value) scores B0 B0
[4]
2.(a)½B1
(b)600 × ½ = 300M1 A1
300/600 s.cB1
[3]
3.(a)3, 3, 5, 3, 3, 3M1
Accept correct tallies
A1
Or 0.15, 0.15, 0.25, 0.15, 0.15, 0.15
At least 3 correct scores SC1
(b)60 × M1
Or scaling up eg 5 × 3
= 15A1 ft
15/60 scores M1A0
15 out of 60 scores M1A1
(c)More than expected with a suitable qualification or only expect 10B1
A correct comparison made. Not 9/60.
[5]
4.(a)40B1
sight of 40: e.g. accept 40/200, 40 out of 200
(b)0.008 (or 1/125) (or 0.8%)M1
1/5 × 1/5 × 1/5, ?/125 implies M1
0.008, 1/125A1
s.c.allow 6 × 1/125 = 6/125
and 3/5 × 2/5 × 1/5 = 6/125
NB: 3/125 is A0
(c)No (can be implied) with reasonB1
for mentioning or implying that 152 is more than is expected
for mentioning that it should be 100 out of 500 or 1/5 probability;B1
minimum for B1B1: no it should be about 100/500 not 152
can use 17 out of 50 spins for reasoning but not results for 5 or 10 spins
NB: yes get B0B0 even if reasoning seems valid
NB: if totalled 173/565, 113/565
[5]
5.(a)0.3, 0.16, 0.23, 0.2, 0.19 plotted on graphB3
or 3 correct values plottedB2
or 1 correct value plottedB1
or 2 correct values seen
(accept fractions or %)B1
Allow joined points or vertical line graph
Bars drawndeduct B1
(b)200B1
Allow 190 to 210
[4]
6.(a)14M1,A1
M1 for 0.28 × 50
A1 cao
(b)0.3B1
(c)biased, should be nearer 0.5B1
[4]