Finding minimum n for P(X=0) threshold

Questions requiring the smallest n such that the probability of zero occurrences falls below a given threshold (e.g., P(X=0) < 0.05 or P(X=0) < 0.01).

7 questions

CAIE S2 2011 June Q4
4 On average, 1 in 2500 people have a particular gene.
  1. Use a suitable approximation to find the probability that, in a random sample of 10000 people, more than 3 people have this gene.
  2. The probability that, in a random sample of \(n\) people, none of them has the gene is less than 0.01 . Find the smallest possible value of \(n\).
CAIE S2 Specimen Q5
5 On average, 1 in 2500 adults has a certain medical condition.
  1. Use a suitable approximation to find the probability that, in a random sample of 4000 people, more than 3 have this condition.
  2. In a random sample of \(n\) people, where \(n\) is large, the probability that none has the condition is less than 0.05 . Find the smallest possible value of \(n\).
CAIE S2 2004 November Q5
5 Of people who wear contact lenses, 1 in 1500 on average have laser treatment for short sight.
  1. Use a suitable approximation to find the probability that, of a random sample of 2700 contact lens wearers, more than 2 people have laser treatment.
  2. In a random sample of \(n\) contact lens wearers the probability that no one has laser treatment is less than 0.01 . Find the least possible value of \(n\).
CAIE S2 2015 November Q5
5 On average, 1 in 2500 adults has a certain medical condition.
  1. Use a suitable approximation to find the probability that, in a random sample of 4000 people, more than 3 have this condition.
  2. In a random sample of \(n\) people, where \(n\) is large, the probability that none has the condition is less than 0.05 . Find the smallest possible value of \(n\).
Edexcel S2 2018 October Q7
7. Members of a conservation group record the number of sightings of a rare animal. The number of sightings follows a Poisson distribution with a rate of 1 every 2 months.
  1. Find the smallest value of \(n\) such that the probability that there are at least \(n\) sightings in 2 months is less than 0.05
  2. Find the smallest number of months, \(m\), such that the probability of no sightings in \(m\) months is less than 0.05
  3. Find the probability that there is at least 1 sighting per month in each of 3 consecutive months.
  4. Find the probability that the number of sightings in an 8 month period is equal to the expected number of sightings for that period.
  5. Given that there were 4 sightings in a 4 month period, find the probability that there were more sightings in the last 2 months than in the first 2 months.
Edexcel S2 2021 October Q4
  1. The number of cars entering a safari park per 10 -minute period can be modelled by a Poisson distribution with mean 6
    1. Find the probability that in a given 10 -minute period exactly 8 cars will enter the safari park.
    2. Find the smallest value of \(n\) such that the probability that at least \(n\) cars enter the safari park in 10 minutes is less than 0.05
    The probability that no cars enter the safari park in \(m\) minutes, where \(m\) is an integer, is less than 0.05
  2. Find the smallest value of \(m\) A car enters the safari park.
  3. Find the probability that there is less than 5 minutes before the next car enters the safari park. Given that exactly 15 cars entered the safari park in a 30-minute period,
  4. find the probability that exactly 1 car entered the safari park in the first 5 minutes of the 30-minute period. Aston claims that the mean number of cars entering the safari park per 10-minute period is more than 6 He selects a 15-minute period at random in order to test whether there is evidence to support his claim.
  5. Determine the critical region for the test at the \(5 \%\) level of significance.
Edexcel FS1 AS 2022 June Q2
  1. Xena catches fish at random, at a constant rate of 0.6 per hour.
    1. Find the probability that Xena catches exactly 4 fish in a 5 -hour period.
    The probability of Xena catching no fish in a period of \(t\) hours is less than 0.16
  2. Find the minimum value of \(t\), giving your answer to one decimal place. Independently of Xena, Zion catches fish at random with a mean rate of 0.8 per hour.
    Xena and Zion try using new bait to catch fish. The number of fish caught in total by Xena and Zion after using the new bait, in a randomly selected 4-hour period, is 12
  3. Use a suitable test to determine, at the \(5 \%\) level of significance, whether or not there is evidence that the rate at which fish are caught has increased after using the new bait. State your hypotheses clearly and the \(p\)-value used in your test.