Teach Me Maths

Relative Frequency and Expected Outcomes

Question 11 mark

Mia spins a spinner 50 times. It lands on red 18 times.

Work out an estimate for the probability that the spinner will land on red the next time Mia spins it.

Hint

Use the fraction of the spins that landed on red.

Worked solution

Mia spins a spinner 50 times. It lands on red 18 times.

Work out an estimate for the probability that the spinner will land on red the next time Mia spins it.

  1. The estimate is the relative frequency: number of reds \div number of spins.
  2. \frac{18}{50}=0.36.

Question 22 marks

Joe drops a drawing pin 40 times. It lands point up 26 times. It lands point down the other times.

Work out an estimate for the probability that the pin will land point down the next time Joe drops it.

Hint

First work out how many times the pin landed point down.

Worked solution

Joe drops a drawing pin 40 times. It lands point up 26 times. It lands point down the other times.

Work out an estimate for the probability that the pin will land point down the next time Joe drops it.

  1. Point down: 40-26=14 times.
  2. Estimate: \frac{14}{40}=0.35.

Question 32 marks

Kim rolls a dice a number of times. The table shows her results.

Work out an estimate for the probability that the dice will land on 6 the next time Kim rolls it.

Hint

Add up the frequencies to find how many times Kim rolled the dice.

Worked solution

Kim rolls a dice a number of times. The table shows her results.

Work out an estimate for the probability that the dice will land on 6 the next time Kim rolls it.

  1. Total rolls: 8+11+9+10+7+15=60.
  2. The dice landed on 6 fifteen times.
  3. Estimate: \frac{15}{60}=\frac{1}{4}=0.25.

Question 42 marks

The probability that a biased coin will land on heads is 0.7

Lena is going to throw the coin 200 times.

Work out an estimate for the number of times the coin will land on heads.

Hint

Multiply the probability by the number of throws.

Worked solution

The probability that a biased coin will land on heads is 0.7

Lena is going to throw the coin 200 times.

Work out an estimate for the number of times the coin will land on heads.

  1. Expected number = probability \times number of throws.
  2. 0.7\times200=140.

Question 51 mark

Sam is going to roll a fair 6-sided dice 90 times.

Work out an estimate for the number of times the dice will land on 4.

Hint

On a fair dice each number is equally likely.

Worked solution

Sam is going to roll a fair 6-sided dice 90 times.

Work out an estimate for the number of times the dice will land on 4.

  1. The probability of landing on 4 is \frac{1}{6}.
  2. \frac{1}{6} of 90 is 90\div6=15.

Question 62 marks

The probability that a seed will grow into a plant is \frac{4}{5}

Mia plants 60 of these seeds.

Work out an estimate for the number of seeds that will grow into a plant.

Hint

Find \frac{4}{5} of 60.

Worked solution

The probability that a seed will grow into a plant is \frac{4}{5}

Mia plants 60 of these seeds.

Work out an estimate for the number of seeds that will grow into a plant.

  1. Estimate =\frac{4}{5}\times60.
  2. 60\div5=12, and 12\times4=48.

Question 72 marks

A factory makes light bulbs. 3\% of the light bulbs it makes are faulty.

One day the factory makes 800 light bulbs.

Work out an estimate for the number of these light bulbs that are faulty.

Hint

Find 1% of 800 first.

Worked solution

A factory makes light bulbs. 3\% of the light bulbs it makes are faulty.

One day the factory makes 800 light bulbs.

Work out an estimate for the number of these light bulbs that are faulty.

  1. 1\% of 800 is 8.
  2. 3\% of 800 is 3\times8=24.

Question 81 mark

Leah spins a spinner 20 times. It lands on blue 9 times.

Omar spins the same spinner 200 times. It lands on blue 62 times.

Whose results will give the better estimate for the probability that the spinner lands on blue?

Hint

Think about which experiment had more trials.

Worked solution

Leah spins a spinner 20 times. It lands on blue 9 times.

Omar spins the same spinner 200 times. It lands on blue 62 times.

Whose results will give the better estimate for the probability that the spinner lands on blue?

  1. Leah's, because a bigger fraction of her spins landed on blue.
  2. Leah's, because fewer spins means fewer mistakes.
  3. Omar's, because he spun the spinner more times.
  4. They are equally good, because they used the same spinner.
  1. The more times an experiment is done, the more reliable the estimate is.
  2. Omar spun the spinner 200 times and Leah spun it only 20 times, so Omar's results are better.

Question 92 marks

A spinner is biased. The table shows the probability that the spinner will land on each colour.

Jake is going to spin the spinner 250 times.

Work out an estimate for the number of times the spinner will land on green.

Hint

Multiply the probability for green by the number of spins.

Worked solution

A spinner is biased. The table shows the probability that the spinner will land on each colour.

Jake is going to spin the spinner 250 times.

Work out an estimate for the number of times the spinner will land on green.

  1. The probability of green is 0.18
  2. Estimate: 0.18\times250=45.

Question 101 mark

Sam is going to throw a fair coin 100 times.

Sam says, "The coin will land on heads exactly 50 times."

Which statement is true?

Hint

Think about whether the results of throwing a coin can be known for certain.

Worked solution

Sam is going to throw a fair coin 100 times.

Sam says, "The coin will land on heads exactly 50 times."

Which statement is true?

  1. Sam is right, because the probability of heads is \frac{1}{2}
  2. 50 is only an estimate. The coin could land on heads a different number of times.
  3. The coin will land on heads fewer than 50 times.
  4. The coin must land on heads at least 50 times.
  1. \frac{1}{2}\times100=50 is the expected number of heads.
  2. This is only an estimate: each throw is random, so the number of heads could be 47, 53 or another number.

Question 112 marks

When a paper cup is dropped, it can land on its side or land upright.

Ben drops the cup 40 times. It lands on its side 26 times.

Cara drops the same cup 60 times. It lands on its side 33 times.

Use all their results to work out an estimate for the probability that the cup lands on its side.

Hint

Add up all the drops and all the times it landed on its side.

Worked solution

When a paper cup is dropped, it can land on its side or land upright.

Ben drops the cup 40 times. It lands on its side 26 times.

Cara drops the same cup 60 times. It lands on its side 33 times.

Use all their results to work out an estimate for the probability that the cup lands on its side.

  1. Total drops: 40+60=100.
  2. Total on its side: 26+33=59.
  3. Estimate: \frac{59}{100}=0.59.

Question 123 marks

Lucy records the colour of 150 cars that drive past her school. The table shows her results, but the number of silver cars is missing.

Work out an estimate for the probability that the next car to drive past is silver.

Hint

Work out the number of silver cars first.

Worked solution

Lucy records the colour of 150 cars that drive past her school. The table shows her results, but the number of silver cars is missing.

Work out an estimate for the probability that the next car to drive past is silver.

  1. Cars that are not silver: 27+41+34+15=117.
  2. Silver cars: 150-117=33.
  3. Estimate: \frac{33}{150}=0.22.

Question 132 marks

The probability that a biased dice will land on 6 is 0.15

Rhys is going to roll the dice a number of times. He works out that an estimate for the number of times the dice will land on 6 is 24

How many times is Rhys going to roll the dice?

Hint

Probability × number of rolls = 24. Work backwards.

Worked solution

The probability that a biased dice will land on 6 is 0.15

Rhys is going to roll the dice a number of times. He works out that an estimate for the number of times the dice will land on 6 is 24

How many times is Rhys going to roll the dice?

  1. 0.15\times number of rolls =24.
  2. Number of rolls =24\div0.15=160.
  3. Check: 0.15\times160=24.

Question 142 marks

The probability that a biased coin lands on tails is 0.35

Jo is going to throw the coin 80 times. Kai is going to throw the coin 140 times.

Work out an estimate for how many more times the coin will land on tails for Kai than for Jo.

Hint

Work out an estimate for each person, then find the difference.

Worked solution

The probability that a biased coin lands on tails is 0.35

Jo is going to throw the coin 80 times. Kai is going to throw the coin 140 times.

Work out an estimate for how many more times the coin will land on tails for Kai than for Jo.

  1. Jo: 0.35\times80=28.
  2. Kai: 0.35\times140=49.
  3. Difference: 49-28=21.

Question 152 marks

There are 840 students at a school.

Ella asks a random sample of 60 of these students how they travel to school. 15 of them say they cycle.

Work out an estimate for the number of students at the school who cycle.

Hint

Find the fraction of the sample who cycle, then use the same fraction of the whole school.

Worked solution

There are 840 students at a school.

Ella asks a random sample of 60 of these students how they travel to school. 15 of them say they cycle.

Work out an estimate for the number of students at the school who cycle.

  1. Fraction of the sample: \frac{15}{60}=\frac{1}{4}.
  2. Estimate: \frac{1}{4} of 840 is 840\div4=210.

Question 16Challenge3 marks

Amy rolls a biased dice 20 times. It lands on 6 eight times.

Ben rolls the same dice 80 times. It lands on 6 twelve times.

Chloe works out \frac{8}{20}=0.4 and \frac{12}{80}=0.15

She then finds the number halfway between 0.4 and 0.15, and says this is the best estimate for the probability that the dice lands on 6

(a)

Use all the results to work out a better estimate for the probability that the dice lands on 6

2 marks

(b)

What is wrong with Chloe's method?

1 mark

Hint

Add the numbers of sixes together and add the numbers of rolls together.

Worked solution

Amy rolls a biased dice 20 times. It lands on 6 eight times.

Ben rolls the same dice 80 times. It lands on 6 twelve times.

Chloe works out \frac{8}{20}=0.4 and \frac{12}{80}=0.15

She then finds the number halfway between 0.4 and 0.15, and says this is the best estimate for the probability that the dice lands on 6

Part (a)

Use all the results to work out a better estimate for the probability that the dice lands on 6

  1. Total rolls: 20+80=100.
  2. Total sixes: 8+12=20.
  3. Estimate: \frac{20}{100}=0.2.

Part (b)

What is wrong with Chloe's method?

  1. Probabilities should be multiplied, not added.
  2. Amy's result should count for more, because she rolled fewer times.
  3. It counts both results equally, but Ben rolled the dice four times as often.
  4. Nothing. Her method always gives the same answer as using all the results.
  1. Chloe's estimate is 0.275, which is not the same as 0.2.
  2. Ben's 80 rolls should count for more than Amy's 20 rolls.
  3. Adding all the results together does this, but finding the number halfway does not.

Question 17Challenge3 marks

Kofi drops a slice of toast 45 times. It lands butter side down 28 times.

Tara is going to drop a slice of toast 270 times.

Use Kofi's results to work out an estimate for the number of times the toast will land butter side down.

Hint

Keep the fraction exact. If you round it to a decimal first, your answer will be wrong.

Worked solution

Kofi drops a slice of toast 45 times. It lands butter side down 28 times.

Tara is going to drop a slice of toast 270 times.

Use Kofi's results to work out an estimate for the number of times the toast will land butter side down.

  1. The relative frequency is \frac{28}{45}. Do not round it.
  2. Estimate: \frac{28}{45}\times270.
  3. 270\div45=6, and 28\times6=168.
  4. Rounding first, for example 0.62\times270=167.4, gives an answer that is not exact.

Question 18Challenge3 marks

Raj spins a biased 5-sided spinner 80 times. The table shows his results.

Raj is going to spin the spinner 400 more times.

Use his results to work out an estimate for the number of times the spinner will land on an even number.

Hint

Find the relative frequency of an even number in Raj's 80 spins.

Worked solution

Raj spins a biased 5-sided spinner 80 times. The table shows his results.

Raj is going to spin the spinner 400 more times.

Use his results to work out an estimate for the number of times the spinner will land on an even number.

  1. Even numbers: 2 and 4. Frequency: 17+19=36.
  2. Relative frequency: \frac{36}{80}=0.45.
  3. Estimate: 0.45\times400=180.

Question 19Challenge4 marks

A spinner is biased. It can land on red, blue or green.

The probability that it lands on blue is 0.25

The probability that it lands on green is 0.43

Hassan is going to spin the spinner 250 times.

(a)

Work out an estimate for the number of times the spinner will land on red.

3 marks

(b)

Hassan spins the spinner 250 times. It lands on red 74 times.

He says, "This proves that the probability of red is not 0.32"

Which statement is true?

1 mark

Hint

For (a), the three probabilities add up to 1.

Worked solution

A spinner is biased. It can land on red, blue or green.

The probability that it lands on blue is 0.25

The probability that it lands on green is 0.43

Hassan is going to spin the spinner 250 times.

Part (a)

Work out an estimate for the number of times the spinner will land on red.

  1. Red: 1-(0.25+0.43)=1-0.68=0.32.
  2. Estimate: 0.32\times250=80.

Part (b)

Hassan spins the spinner 250 times. It lands on red 74 times.

He says, "This proves that the probability of red is not 0.32"

Which statement is true?

  1. Hassan is right, because 74 is not the same as his estimate.
  2. Hassan is right, because the spinner should land on red exactly as many times as his estimate.
  3. Hassan is wrong, because the spinner must land on red fewer times than his estimate.
  4. Hassan is wrong. His answer to (a) is only an estimate, and 74 is close to it.
  1. The spins are random, so the number of reds will not always be exactly 80.
  2. 74 is close to the estimate of 80, so these results do not prove the probability is wrong.

Question 20Challenge4 marks

Kerry runs a game at a school fair. Each player pays 50p to play. A player who wins the game gets a prize of £2

Before the fair, Kerry plays the game 40 times. She wins 6 times.

Kerry expects 120 people to play the game at the fair.

Use Kerry's results to work out an estimate for the profit she will make from the game.

Hint

Estimate how many of the 120 players will win a prize, then compare the money paid in with the money paid out.

Worked solution

Kerry runs a game at a school fair. Each player pays 50p to play. A player who wins the game gets a prize of £2

Before the fair, Kerry plays the game 40 times. She wins 6 times.

Kerry expects 120 people to play the game at the fair.

Use Kerry's results to work out an estimate for the profit she will make from the game.

  1. Relative frequency of a win: \frac{6}{40}=0.15.
  2. Estimated number of winners: 0.15\times120=18.
  3. Money paid out in prizes: 18\times2=36, so £36.
  4. Money paid in: 120\times0.50=60, so £60.
  5. Profit: 60-36=24, so £24.

More on this topic: Relative Frequency and Expected Outcomes worksheet with full solutions

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