Questions — Edexcel FS1 (49 questions)

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Edexcel FS1 2019 June Q1
  1. A chocolate manufacturer places special tokens in \(2 \%\) of the bars it produces so that each bar contains at most one token. Anyone who collects 3 of these tokens can claim a prize.
Andreia buys a box of 40 bars of the chocolate.
  1. Find the probability that Andreia can claim a prize. Barney intends to buy bars of the chocolate, one at a time, until he can claim a prize.
  2. Find the probability that Barney can claim a prize when he buys his 40th bar of chocolate.
  3. Find the expected number of bars that Barney must buy to claim a prize.
Edexcel FS1 2019 June Q2
  1. Indre works on reception in an office and deals with all the telephone calls that arrive. Calls arrive randomly and, in a 4-hour morning shift, there are on average 80 calls.
    1. Using a suitable model, find the probability of more than 4 calls arriving in a particular 20 -minute period one morning.
    Indre is allowed 20 minutes of break time during each 4-hour morning shift, which she can take in 5 -minute periods. When she takes a break, a machine records details of any call in the office that Indre has missed. One morning Indre took her break time in 4 periods of 5 minutes each.
  2. Find the probability that in exactly 3 of these periods there were no calls. On another occasion Indre took 1 break of 5 minutes and 1 break of 15 minutes.
  3. Find the probability that Indre missed exactly 1 call in each of these 2 breaks.
Edexcel FS1 2019 June Q3
  1. A biased spinner can land on the numbers \(1,2,3,4\) or 5 with the following probabilities.
Number on spinner12345
Probability0.30.10.20.10.3
The spinner will be spun 80 times and the mean of the numbers it lands on will be calculated. Find an estimate of the probability that this mean will be greater than 3.25
(6)
Edexcel FS1 2019 June Q4
  1. Liam and Simone are studying the distribution of oak trees in some woodland. They divided the woodland into 80 equal squares and recorded the number of oak trees in each square. The results are summarised in Table 1 below.
\begin{table}[h]
Number of oak trees in a square01234567 or more
Frequency142123131170
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table} Liam believes that the oak trees were deliberately planted, with 6 oak trees per square and that a constant proportion \(p\) of the oak trees survived.
  1. Suggest the model Liam should use to describe the number of oak trees per square. Liam decides to test whether or not his model is suitable and calculates the expected frequencies given in Table 2. \begin{table}[h]
    Number of oak trees in a square0 or 123456
    Expected frequency5.5314.8924.2622.2410.872.21
    \captionsetup{labelformat=empty} \caption{Table 2}
    \end{table}
  2. Showing your working clearly, complete the test using a \(5 \%\) level of significance. You should state your critical value and conclusion clearly. Simone believes that a Poisson distribution could be used to model the number of oak trees per square. She calculates the expected frequencies given in Table 3. \begin{table}[h]
    Number of oak trees in a square0 or 123456 or more
    Expected frequency12.6916.07\(s\)14.58\(t\)9.37
    \captionsetup{labelformat=empty} \caption{Table 3}
    \end{table}
  3. Find the value of \(s\) and the value of \(t\), giving your answers to 2 decimal places.
  4. Write down hypotheses to test the suitability of Simone's model. The test statistic for this test is 8.749
  5. Complete the test. Use a \(5 \%\) level of significance and state your critical value and conclusion clearly.
  6. Using the results of these tests, explain whether the origin of this woodland is likely to be cultivated or wild.
Edexcel FS1 2019 June Q5
  1. Information was collected about accidents on the Seapron bypass. It was found that the number of accidents per month could be modelled by a Poisson distribution with mean 2.5 Following some work on the bypass, the numbers of accidents during a series of 3-month periods were recorded. The data were used to test whether or not there was a change in the mean number of accidents per month.
    1. Stating your hypotheses clearly and using a \(5 \%\) level of significance, find the critical region for this test. You should state the probability in each tail.
    2. State P(Type I error) using this test.
    Data from the series of 3-month periods are recorded for 2 years.
  2. Find the probability that at least 2 of these 3-month periods give a significant result. Given that the number of accidents per month on the bypass, after the work is completed, is actually 2.1 per month,
  3. find P (Type II error) for the test in part (a)
Edexcel FS1 2019 June Q6
  1. The discrete random variable \(X\) has probability generating function
$$\mathrm { G } _ { X } ( t ) = k \ln \left( \frac { 2 } { 2 - t } \right)$$ where \(k\) is a constant.
  1. Find the exact value of \(k\)
  2. Find the exact value of \(\operatorname { Var } ( X )\)
  3. Find \(\mathrm { P } ( X = 3 )\)
Edexcel FS1 2019 June Q7
  1. A spinner can land on red or blue. When the spinner is spun, there is a probability of \(\frac { 1 } { 3 }\) that it lands on blue. The spinner is spun repeatedly.
The random variable \(B\) represents the number of the spin when the spinner first lands on blue.
  1. Find (i) \(\mathrm { P } ( B = 4 )\)
    (ii) \(\mathrm { P } ( B \leqslant 5 )\)
  2. Find \(\mathrm { E } \left( B ^ { 2 } \right)\) Steve invites Tamara to play a game with this spinner.
    Tamara must choose a colour, either red or blue.
    Steve will spin the spinner repeatedly until the spinner first lands on the colour Tamara has chosen. The random variable \(X\) represents the number of the spin when this occurs. If Tamara chooses red, her score is \(\mathrm { e } ^ { X }\)
    If Tamara chooses blue, her score is \(X ^ { 2 }\)
  3. State, giving your reasons and showing any calculations you have made, which colour you would recommend that Tamara chooses.
Edexcel FS1 2020 June Q1
  1. The number of customers entering Jeff's supermarket each morning follows a Poisson distribution.
Past information shows that customers enter at an average rate of 2 every 5 minutes.
Using this information,
    1. find the probability that exactly 26 customers enter Jeff's supermarket during a randomly selected 1-hour period one morning,
    2. find the probability that at least 21 customers enter Jeff's supermarket during a randomly selected 1-hour period one morning. A rival supermarket is opened nearby. Following its opening, the number of customers entering Jeff's supermarket over a randomly selected 40-minute period is found to be 10
  2. Test, at the 5\% significance level, whether or not there is evidence of a decrease in the rate of customers entering Jeff's supermarket. State your hypotheses clearly. A further randomly selected 20 -minute period is observed and the hypothesis test is repeated. Given that the true rate of customers entering Jeff's supermarket is now 1 every 5 minutes,
  3. calculate the probability of a Type II error.
Edexcel FS1 2020 June Q2
  1. The discrete random variables \(W , X\) and \(Y\) are distributed as follows
$$W \sim \mathrm {~B} ( 10,0.4 ) \quad X \sim \operatorname { Po } ( 4 ) \quad Y \sim \operatorname { Po } ( 3 )$$
  1. Explain whether or not \(\mathrm { Po } ( 4 )\) would be a good approximation to \(\mathrm { B } ( 10,0.4 )\)
  2. State the assumption required for \(X + Y\) to be distributed as \(\operatorname { Po } ( 7 )\) Given the assumption in part (b) holds,
  3. find \(\mathrm { P } ( X + Y < \operatorname { Var } ( W ) )\)
Edexcel FS1 2020 June Q3
  1. Suzanne and Jon are playing a game.
They put 4 red counters and 1 blue counter in a bag.
Suzanne reaches into the bag and selects one of the counters at random. If the counter she selects is blue, she wins the game. Otherwise she puts it back in the bag and Jon selects one at random. If the counter he selects is blue, he wins the game. Otherwise he puts it back in the bag and they repeat this process until one of them selects the blue counter.
  1. Find the probability that Suzanne selects the blue counter on her 4th selection.
  2. Find the probability that the blue counter is first selected on or after Jon's third selection.
  3. Find the mean and standard deviation of the number of selections made until the blue counter is selected.
  4. Find the probability that Suzanne wins the game.
Edexcel FS1 2020 June Q4
  1. The discrete random variable \(X\) has the following probability distribution.
\(x\)- 5- 234
\(\mathrm { P } ( X = x )\)\(\frac { 1 } { 12 }\)\(\frac { 1 } { 6 }\)\(\frac { 1 } { 4 }\)\(\frac { 1 } { 2 }\)
  1. Find \(\operatorname { Var } ( X )\) The discrete random variable \(Y\) is defined in terms of the discrete random variable \(X\)
    When \(X\) is negative, \(Y = X ^ { 2 }\)
    When \(X\) is positive, \(Y = 3 X - 2\)
  2. Find \(\mathrm { P } ( Y < 9 )\)
  3. Find \(\mathrm { E } ( X Y )\)
Edexcel FS1 2020 June Q5
  1. A factory produces pins.
An engineer selects 40 independent random samples of 6 pins produced at the factory and records the number of defective pins in each sample.
Number of defective pins0123456
Observed frequency191172010
  1. Show that the proportion of defective pins in the 40 samples is 0.15 The engineer suggests that the number of defective pins in a sample of 6 can be modelled using a binomial distribution. Using the information from the sample above, a test is to be carried out at the \(10 \%\) significance level, to see whether the data are consistent with the engineer's suggested model. The value of the test statistic for this test is 2.689
  2. Justifying the degrees of freedom used, carry out the test, at the \(10 \%\) significance level, to see whether the data are consistent with the engineer's suggested model. State your hypotheses clearly. The engineer later discovers that the previously recorded information was incorrect. The data should have been as follows.
    Number of defective pins0123456
    Observed frequency191163100
  3. Describe the effect this would have on the value of the test statistic that should be used for the hypothesis test.
    Give reasons for your answer.
Edexcel FS1 2020 June Q6
  1. A discrete random variable \(X\) has probability generating function given by
$$\mathrm { G } _ { X } ( t ) = \frac { 1 } { 64 } \left( a + b t ^ { 2 } \right) ^ { 2 }$$ where \(a\) and \(b\) are positive constants.
  1. Write down the value of \(\mathrm { P } ( X = 3 )\) Given that \(\mathrm { P } ( X = 4 ) = \frac { 25 } { 64 }\)
    1. find \(\mathrm { P } ( X = 2 )\)
    2. find \(\mathrm { E } ( X )\) The random variable \(Y = 3 X + 2\)
  3. Find the probability generating function of \(Y\)
Edexcel FS1 2020 June Q7
  1. A six-sided die has sides labelled \(1,2,3,4,5\) and 6
The random variable \(S\) represents the score when the die is rolled.
Alicia rolls the die 45 times and the mean score, \(\bar { S }\), is calculated.
Assuming the die is fair and using a suitable approximation,
  1. find, to 3 significant figures, the value of \(k\) such that \(\mathrm { P } ( \bar { S } < k ) = 0.05\)
  2. Explain the relevance of the Central Limit Theorem in part (a). Alicia considers the following hypotheses:
    \(\mathrm { H } _ { 0 }\) : The die is fair
    \(\mathrm { H } _ { 1 }\) : The die is not fair
    If \(\bar { S } < 3.1\) or \(\bar { S } > 3.9\), then \(\mathrm { H } _ { 0 }\) will be rejected.
    Given that the true distribution of \(S\) has mean 4 and variance 3
  3. find the power of this test.
  4. Describe what would happen to the power of this test if Alicia were to increase the number of rolls of the die.
    Give a reason for your answer.
Edexcel FS1 2021 June Q1
  1. Kelly throws a tetrahedral die \(n\) times and records the number on which it lands for each throw.
She calculates the expected frequency for each number to be 43 if the die was unbiased.
The table below shows three of the frequencies Kelly records but the fourth one is missing.
Number1234
Frequency473436\(x\)
  1. Show that \(x = 55\) Kelly wishes to test, at the \(5 \%\) level of significance, whether or not there is evidence that the tetrahedral die is unbiased.
  2. Explain why there are 3 degrees of freedom for this test.
  3. Stating your hypotheses clearly and the critical value used, carry out the test.
Edexcel FS1 2021 June Q2
  1. On a weekday, a garage receives telephone calls randomly, at a mean rate of 1.25 per 10 minutes.
    1. Show that the probability that on a weekday at least 2 calls are received by the garage in a 30 -minute period is 0.888 to 3 decimal places.
    2. Calculate the probability that at least 2 calls are received by the garage in fewer than 4 out of 6 randomly selected, non-overlapping 30-minute periods on a weekday.
    The manager of the garage randomly selects 150 non-overlapping 30-minute periods on weekdays.
    She records the number of calls received in each of these 30-minute periods.
  2. Using a Poisson approximation show that the probability of the manager finding at least 3 of these 30 -minute periods when exactly 8 calls are received by the garage is 0.664 to 3 significant figures.
  3. Explain why the Poisson approximation may be reasonable in this case. The manager of the garage decides to test whether the number of calls received on a Saturday is different from the number of calls received on a weekday. She selects a Saturday at random and records the number of telephone calls received by the garage in the first 4 hours.
  4. Write down the hypotheses for this test. The manager found that there had been 40 telephone calls received by the garage in the first 4 hours.
  5. Carry out the test using a \(5 \%\) level of significance.
Edexcel FS1 2021 June Q3
  1. A courier delivers parcels. The random variable \(X\) represents the number of parcels delivered successfully each day by the courier where \(X \sim \mathrm {~B} ( 400,0.64 )\)
A random sample \(X _ { 1 } , X _ { 2 } , \ldots X _ { 100 }\) is taken.
Estimate the probability that the mean number of parcels delivered each day by the courier is greater than 257
Edexcel FS1 2021 June Q4
  1. Members of a photographic group may enter a maximum of 5 photographs into a members only competition.
    Past experience has shown that the number of photographs, \(N\), entered by a member follows the probability distribution shown below.
\(n\)012345
\(\mathrm { P } ( N = n )\)\(a\)0.20.050.25\(b\)\(c\)
Given that \(\mathrm { E } ( 4 N + 2 ) = 14.8\) and \(\mathrm { P } ( N = 5 \mid N > 2 ) = \frac { 1 } { 2 }\)
  1. show that \(\operatorname { Var } ( N ) = 2.76\) The group decided to charge a 50p entry fee for the first photograph entered and then 20p for each extra photograph entered into the competition up to a maximum of \(\pounds 1\) per person. Thus a member who enters 3 photographs pays 90 p and a member who enters 4 or 5 photographs just pays £1 Assuming that the probability distribution for the number of photographs entered by a member is unchanged,
  2. calculate the expected entry fee per member. Bai suggests that, as the mean and variance are close, a Poisson distribution could be used to model the number of photographs entered by a member next year.
  3. State a limitation of the Poisson distribution in this case.
Edexcel FS1 2021 June Q5
  1. Asha, Davinda and Jerry each have a bag containing a large number of counters, some of which are white and the rest are red.
    Each person draws counters from their bag one at a time, notes the colour of the counter and returns it to their bag.
The probability of Asha getting a red counter on any one draw is 0.07
  1. Find the probability that Asha will draw at least 3 white counters before a red counter is drawn.
  2. Find the probability that Asha gets a red counter for the second time on her 9th draw. The probability of Davinda getting a red counter on any one draw is \(p\). Davinda draws counters until she gets \(n\) red counters. The random variable \(D\) is the number of counters Davinda draws. Given that the mean and the standard deviation of \(D\) are 4400 and 660 respectively,
  3. find the value of \(p\). Jerry believes that his bag contains a smaller proportion of red counters than Asha's bag. To test his belief, Jerry draws counters from his bag until he gets a red counter. Jerry defines the random variable \(J\) to be the number of counters drawn up to and including the first red counter.
  4. Stating your hypotheses clearly and using a \(10 \%\) level of significance, find the critical region for this test. Jerry gets a red counter for the first time on his 34th draw.
  5. Giving a reason for your answer, state whether or not there is evidence that Jerry's bag contains a smaller proportion of red counters than Asha’s bag. Given that the probability of Jerry getting a red counter on any one draw is 0.011
  6. show that the power of the test is 0.702 to 3 significant figures.
Edexcel FS1 2021 June Q6
  1. The probability generating function of the random variable \(X\) is
$$\mathrm { G } _ { X } ( t ) = k ( 1 + 2 t ) ^ { 5 }$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { 243 }\)
  2. Find \(\mathrm { P } ( X = 2 )\)
  3. Find the probability generating function of \(W = 2 X + 3\) The probability generating function of the random variable \(Y\) is $$\mathrm { G } _ { Y } ( t ) = \frac { t ( 1 + 2 t ) ^ { 2 } } { 9 }$$ Given that \(X\) and \(Y\) are independent,
  4. find the probability generating function of \(U = X + Y\) in its simplest form.
  5. Use calculus to find the value of \(\operatorname { Var } ( U )\)
Edexcel FS1 2021 June Q7
  1. A manufacturer has a machine that produces lollipop sticks.
The length of a lollipop stick produced by the machine is normally distributed with unknown mean \(\mu\) and standard deviation 0.2 Farhan believes that the machine is not working properly and the mean length of the lollipop sticks has decreased.
He takes a random sample of size \(n\) to test, at the 1\% level of significance, the hypotheses $$\mathrm { H } _ { 0 } : \mu = 15 \quad \mathrm { H } _ { 1 } : \mu < 15$$
  1. Write down the size of this test. Given that the actual value of \(\mu\) is 14.9
    1. calculate the minimum value of \(n\) such that the probability of a Type II error is less than 0.05
      Show your working clearly.
    2. Farhan uses the same sample size, \(n\), but now carries out the test at a \(5 \%\) level of significance. Without doing any further calculations, state how this would affect the probability of a Type II error.
Edexcel FS1 2022 June Q1
  1. A researcher is investigating the number of female cubs present in litters of size 4 He believes that the number of female cubs in a litter can be modelled by \(\mathrm { B } ( 4,0.5 )\) He randomly selects 100 litters each of size 4 and records the number of female cubs. The results are recorded in the table below.
Number of female cubs01234
Observed number of litters103333159
He calculated the expected frequencies as follows
Number of female cubs01234
Expected number of litters6.25\(r\)\(s\)\(r\)6.25
  1. Find the value of \(r\) and the value of \(s\)
  2. Carry out a suitable test, at the \(5 \%\) level of significance, to determine whether or not the number of female cubs in a litter can be modelled by \(\mathrm { B } ( 4,0.5 )\) You should clearly state your hypotheses and the critical value used.
Edexcel FS1 2022 June Q2
  1. The discrete random variable \(X\) has probability distribution
\(x\)- 5- 105\(b\)
\(\mathrm { P } ( X = x )\)0.30.250.10.150.2
where \(b\) is a constant and \(b > 5\)
  1. Find \(\mathrm { E } ( X )\) in terms of \(b\) Given that \(\operatorname { Var } ( X ) = 34.26\)
  2. find the value of \(b\)
  3. Find \(\mathrm { P } \left( X ^ { 2 } < 2 - 3 X \right)\)
Edexcel FS1 2022 June Q3
  1. During the summer, mountain rescue team \(A\) receives calls for help randomly with a rate of 0.4 per day.
    1. Find the probability that during the summer, mountain rescue team \(A\) receives at least 19 calls for help in 28 randomly selected days.
    The leader of mountain rescue team \(A\) randomly selects 250 summer days from the last few years.
    She records the number of calls for help received on each of these days.
  2. Using a Poisson approximation, estimate the probability of the leader finding at least 20 of these days when more than 1 call for help was received by mountain rescue team \(A\). Mountain rescue team \(A\) believes that the number of calls for help per day is lower in the winter than in the summer. The number of calls for help received in 42 randomly selected winter days is 8
  3. Use a suitable test, at the \(5 \%\) level of significance, to assess whether or not there is evidence that the number of calls for help per day is lower in the winter than in the summer. State your hypotheses clearly. During the summer, mountain rescue team \(B\) receives calls for help randomly with a rate of 0.2 per day, independently of calls to mountain rescue team \(A\). The random variable \(C\) is the total number of calls for help received by mountain rescue teams \(A\) and \(B\) during a period of \(n\) days in the summer.
    On a Monday in the summer, mountain rescue teams \(A\) and \(B\) each receive a call for help. Given that over the next \(n\) days \(\mathrm { P } ( C = 0 ) < 0.001\)
  4. calculate the minimum value of \(n\)
  5. Write down an assumption that needs to be made for the model to be appropriate.
Edexcel FS1 2022 June Q4
  1. In a game a spinner is spun repeatedly. When the spinner is spun, the probability of it landing on blue is 0.11
    1. Find the probability that the spinner lands on blue
      1. for the first time on the 6th spin,
      2. for the first time before the 6th spin,
      3. exactly 4 times during the first 6 spins,
      4. for the 4th time on or before the 6th spin.
    Zac and Izana play the game. They take turns to spin the spinner. The winner is the first one to have the spinner land on blue. Izana spins the spinner first.
  2. Show that the probability of Zac winning is 0.471 to 3 significant figures.