KS3 Maths · Year 7 · Chapter 3: Probability · Lesson 3

Estimates of probability

Year 7About 60 minutesNo calculator45 marks

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Date

Do Now

Quick questions to start.

A1
Needed today
Write $\frac{9}{25}$ as a decimal.
[1 mark]
Answer
A2
Needed today
Work out $0.15 \times 400$ .
[1 mark]
Answer
A3
Last lesson
Two fair coins are tossed. Find P(two tails).
[1 mark]
Answer
A4
From chapter 1
Find the LCM of $6$ and $9$ .
[1 mark]
Answer

Example, then your turn

Year 7

Your teacher works through each example with you. Then try the one beside it.

Example 1

A gardener plants $40$ seeds and $34$ of them grow. (a) Estimate the probability that a seed grows. (b) She plants $300$ more seeds. How many would you expect to grow?

Hint An estimate of a probability is the number of times it happened divided by the number of trials.

B1
A shop tests $60$ batteries and $12$ of them fail within a year. (i) Estimate the probability that a battery fails within a year. (ii) How many of $900$ batteries would you expect to fail?
[2 marks]
Answer(i) $\dots\dots$ (ii) $\dots\dots$

Example 2

A dice is rolled $150$ times and lands on $6$ a total of $33$ times. (a) Estimate P( $6$ ). (b) How many $6$ s do you expect in $600$ rolls? (c) How many $6$ s would a fair dice give in $600$ rolls?

Hint For a fair dice, P(

6

)

= \frac{1}{6}

B2
A spinner has four equal sections numbered $1$ to $4$ . In $200$ spins it lands on $3$ a total of $62$ times. (i) Estimate P( $3$ ). (ii) How many $3$ s do you expect in $1000$ spins? (iii) How many $3$ s would a fair spinner give in $1000$ spins?
[3 marks]
Answer(i) $\dots\dots$ (ii) $\dots\dots$ (iii) $\dots\dots$

Example 3

A vet records the injections given to animals over $5$ days. (a) Estimate the probability that an animal needs no injection. (b) The vet works $220$ days a year. Estimate how many animals in a year need more than one injection.

Hint For (b),

18

animals in

5

days is the same rate as

18 \div 5

animals a day.

B3
A library records how $180$ books were returned over $4$ days. (i) Estimate the probability that a book is returned on time. (ii) The library is open $300$ days a year. Estimate how many books a year are returned over a week late.
[3 marks]
Answer(i) $\dots\dots$ (ii) $\dots\dots$

Example 4

Ravi drops a paper cup and records how often it lands on its side. Complete the relative frequencies. Which is the best estimate of the probability that the cup lands on its side?

Hint Relative frequency

=

number of times it happened

\div

number of trials. More trials give a better estimate.

B4
A machine makes items, and some are faulty. Complete the table. Then write down the best estimate of the probability that an item is faulty.
[3 marks]
Answer

Practice

Year 7

C1
A four-sided spinner is spun $80$ times. It lands on $1$ , $2$ , $3$ and $4$ a total of $18$ , $22$ , $17$ and $23$ times. Kim says, "It is biased, because the numbers are not all $20$ ." Is Kim right? Give a reason.
[2 marks]
C2
A bag holds $60$ counters, each red or blue. Leah takes a counter, notes its colour and puts it back. She does this $40$ times and gets red $14$ times. Estimate how many red counters are in the bag.
[2 marks]
Answer

C3
After $20$ games the relative frequency of Mia winning is $0.65$ . After $25$ games it is $0.6$ . How many of the last $5$ games did Mia win?
[3 marks]
Answer
C4
Three pupils do the same experiment. Ann gets $9$ successes in $20$ trials, Ben $22$ in $50$ and Cal $86$ in $200$ . Whose results give the best estimate of P(success)? Give it as a decimal.
[2 marks]
Answer

Exam-style questions

GCSE Foundation

D1
Jo throws a dice $30$ times and gets $9$ fives. She says, "So if I throw it $300$ times, I will get exactly $90$ fives."
Is Jo correct? Explain your answer. How could Jo get a better estimate of the probability of a five?
[3 marks]

D2
Ella spins a spinner. After $10$ , $20$ , $30$ , $40$ and $50$ spins it has landed on blue $2$ , $6$ , $9$ , $13$ and $16$ times.
(a)
The relative frequency of blue after $10$ spins is $0.2$ . Work out the relative frequency after $20$ , $30$ , $40$ and $50$ spins.
[2 marks]
(b)
Plot the relative frequencies on the graph. The first one is done for you.
[2 marks]
(c)
Write down the best estimate of the probability of blue.
[1 mark]
Answer
(d)
Ella spins the spinner $400$ times. Estimate how many times it lands on blue.
[1 mark]
Answer
Total for D2: 6 marks
D3
Ali throws a ball at a basket. The graph shows the relative frequency of scoring a basket after every $10$ throws.
(a)
How many baskets did Ali score in his first $20$ throws?
[1 mark]
Answer
(b)
How many baskets did he score from throw $21$ to throw $40$ ?
[2 marks]
Answer
(c)
Write down the best estimate of the probability that Ali scores a basket.
[1 mark]
Answer
Total for D3: 4 marks
D4
A spinner has five equal sections numbered $1$ to $5$ . It is spun $250$ times. The table shows the results.
(a)
If the spinner is fair, how many times would you expect each number?
[1 mark]
Answer
(b)
Estimate the probability that the spinner lands on $3$ .
[1 mark]
Answer
(c)
Is the spinner fair? Give a reason for your answer.
[2 marks]
Total for D4: 4 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Amir spins a spinner and the relative frequency of red is $0.35$ . He spins it $20$ more times and gets red every time. Now the relative frequency of red is $0.48$ . How many times did he spin it at first?
[4 marks]
Hint 1 · What to try
Try a number of spins at first, for example $40$ , and see what the relative frequency would become.
Hint 2 · The first line
If he spun it $n$ times at first, he had $0.35 \times n$ reds. After the extra spins he has $0.35 n + 20$ reds out of $n + 20$ spins.
Hint 3 · The full method
Try $n = 80$ : $28$ reds at first, then $48$ reds out of $100$ spins, which is $0.48$ . So $80$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can estimate a probability from the results of an experiment.
I can work out relative frequencies and choose the best estimate.
I can use an estimate to say how often something should happen.
I can decide whether results show that something is biased.

Answers and mark scheme

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