Poisson with binomial combination

A question is this type if and only if it involves both Poisson and binomial distributions in the same problem, typically where Poisson events are followed by binomial trials.

10 questions

CAIE S2 2006 June Q6
6 A dressmaker makes dresses for Easifit Fashions. Each dress requires \(2.5 \mathrm {~m} ^ { 2 }\) of material. Faults occur randomly in the material at an average rate of 4.8 per \(20 \mathrm {~m} ^ { 2 }\).
  1. Find the probability that a randomly chosen dress contains at least 2 faults. Each dress has a belt attached to it to make an outfit. Independently of faults in the material, the probability that a belt is faulty is 0.03 . Find the probability that, in an outfit,
  2. neither the dress nor its belt is faulty,
  3. the dress has at least one fault and its belt is faulty. The dressmaker attaches 300 randomly chosen belts to 300 randomly chosen dresses. An outfit in which the dress has at least one fault and its belt is faulty is rejected.
  4. Use a suitable approximation to find the probability that fewer than 3 outfits are rejected.
OCR MEI Paper 2 2019 June Q8
8 A team called "The Educated Guess" enter a weekly quiz. If they win the quiz in a particular week, the probability that they will win the following week is 0.4 , but if they do not win, the probability that they will win the following week is 0.2 . In week 4 The Educated Guess won the quiz.
  1. Calculate the probability that The Educated Guess will win the quiz in week 6. Every week the same 20 quiz teams, each with 6 members, take part in a quiz. Every member of every team buys a raffle ticket. Five winning tickets are drawn randomly, without replacement. Alf, who is a member of one of the teams, takes part every week.
  2. Calculate the probability that, in a randomly chosen week, Alf wins a raffle prize.
  3. Find the smallest number of weeks after which it will be \(95 \%\) certain that Alf has won at least one raffle prize.
Edexcel S2 2023 January Q5
  1. A company produces steel cable.
Defects in the steel cable produced by this company occur at random, at a constant rate of 1 defect per 16 metres. On one day the company produces a piece of steel cable 80 metres long.
  1. Find the probability that there are at most 5 defects in this piece of steel cable. The company produces a piece of steel cable 80 metres long on each of the next 4 days.
  2. Find the probability that fewer than 2 of these 4 pieces of steel cable contain at most 5 defects. The following week the company produces a piece of steel cable \(x\) metres long.
    Using a normal approximation, the probability that this piece of steel cable has fewer than 26 defects is 0.5398
  3. Find the value of \(x\)
Edexcel S2 2022 June Q1
  1. The independent random variables \(W\) and \(X\) have the following distributions.
$$W \sim \operatorname { Po } ( 4 ) \quad X \sim \mathrm {~B} ( 3,0.8 )$$
  1. Write down the value of the variance of \(W\)
  2. Determine the mode of \(X\) Show your working clearly. One observation from each distribution is recorded as \(W _ { 1 }\) and \(X _ { 1 }\) respectively.
  3. Find \(\mathrm { P } \left( W _ { 1 } = 2 \right.\) and \(\left. X _ { 1 } = 2 \right)\)
  4. Find \(\mathrm { P } \left( X _ { 1 } < W _ { 1 } \right)\)
Edexcel S2 2011 June Q5
5. Defects occur at random in planks of wood with a constant rate of 0.5 per 10 cm length. Jim buys a plank of length 100 cm .
  1. Find the probability that Jim's plank contains at most 3 defects. Shivani buys 6 planks each of length 100 cm .
  2. Find the probability that fewer than 2 of Shivani's planks contain at most 3 defects.
  3. Using a suitable approximation, estimate the probability that the total number of defects on Shivani's 6 planks is less than 18.
Edexcel S2 2012 June Q4
4. The number of houses sold by an estate agent follows a Poisson distribution, with a mean of 2 per week.
  1. Find the probability that in the next 4 weeks the estate agent sells,
    1. exactly 3 houses,
    2. more than 5 houses. The estate agent monitors sales in periods of 4 weeks.
  2. Find the probability that in the next twelve of these 4 week periods there are exactly nine periods in which more than 5 houses are sold. The estate agent will receive a bonus if he sells more than 25 houses in the next 10 weeks.
  3. Use a suitable approximation to estimate the probability that the estate agent receives a bonus.
Edexcel S2 2014 June Q3
3. Accidents occur randomly at a road junction at a rate of 18 every year. The random variable \(X\) represents the number of accidents at this road junction in the next 6 months.
  1. Write down the distribution of \(X\).
  2. Find \(\mathrm { P } ( X > 7 )\).
  3. Show that the probability of at least one accident in a randomly selected month is 0.777 (correct to 3 decimal places).
  4. Find the probability that there is at least one accident in exactly 4 of the next 6 months.
Edexcel S2 Q6
6. On a typical weekday morning customers arrive at a village post office independently and at a rate of 3 per 10 minute period. Find the probability that
  1. at least 4 customers arrive in the next 10 minutes,
  2. no more than 7 customers arrive between 11.00 a.m. and 11.30 a.m. The period from 11.00 a.m. to 11.30 a.m. next Tuesday morning will be divided into 6 periods of 5 minutes each.
  3. Find the probability that no customers arrive in at most one of these periods. The post office is open for \(3 \frac { 1 } { 2 }\) hours on Wednesday mornings.
  4. Using a suitable approximation, estimate the probability that more than 49 customers arrive at the post office next Wednesday morning. END
Edexcel S2 Q7
7. In a certain field, daisies are randomly distributed, at an average density of 0.8 daisies per \(\mathrm { cm } ^ { 2 }\). One particular patch, of area \(1 \mathrm {~cm} ^ { 2 }\), is selected at random. Assuming that the number of daisies per \(\mathrm { cm } ^ { 2 }\) has a Poisson distribution,
  1. find the probability that the chosen patch contains
    1. no daisies,
    2. one daisy. Ten such patches are chosen. Using your answers to part (a),
  2. find the probability that the total number of daisies is less than two.
  3. By considering the distribution of daisies over patches of \(10 \mathrm {~cm} ^ { 2 }\), use the Poisson distribution to find the probability that a particular area of \(10 \mathrm {~cm} ^ { 2 }\) contains no more than one daisy.
  4. Compare your answers to parts (b) and (c).
  5. Use a suitable approximation to find the probability that a patch of area \(1 \mathrm {~m} ^ { 2 }\) contains more than 8100 daisies.
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.