Finding maximum n for P(X=0) threshold

Questions requiring the largest n such that the probability of zero occurrences exceeds a given threshold (e.g., P(X=0) > 0.9 or P(X=0) ≥ 0.95).

5 questions

CAIE S2 2021 June Q5
5 On average, 1 in 75000 adults has a certain genetic disorder.
  1. Use a suitable approximating distribution to find the probability that, in a random sample of 10000 people, at least 1 has the genetic disorder.
  2. In a random sample of \(n\) people, where \(n\) is large, the probability that no-one has the genetic disorder is more than 0.9 . Find the largest possible value of \(n\).
CAIE S2 2020 March Q4
4 The number of accidents on a certain road has a Poisson distribution with mean 0.4 per 50-day period.
  1. Find the probability that there will be fewer than 3 accidents during a year (365 days).
  2. The probability that there will be no accidents during a period of \(n\) days is greater than 0.95 . Find the largest possible value of \(n\).
CAIE S2 2003 November Q4
4 The number of emergency telephone calls to the electricity board office in a certain area in \(t\) minutes is known to follow a Poisson distribution with mean \(\frac { 1 } { 80 } t\).
  1. Find the probability that there will be at least 3 emergency telephone calls to the office in any 20-minute period.
  2. The probability that no emergency telephone call is made to the office in a period of \(k\) minutes is 0.9 . Find \(k\).
CAIE S2 2013 November Q4
4 The number of radioactive particles emitted per 150-minute period by some material has a Poisson distribution with mean 0.7.
  1. Find the probability that at most 2 particles will be emitted during a randomly chosen 10 -hour period.
  2. Find, in minutes, the longest time period for which the probability that no particles are emitted is at least 0.99 .
Edexcel S2 2022 June Q5
  1. The number of particles per millilitre in a solution is modelled by a Poisson distribution with mean 0.15
A randomly selected 50 millilitre sample of the solution is taken.
  1. Find the probability that
    1. exactly 10 particles are found,
    2. between 6 and 11 particles (inclusive) are found. Petra takes 12 independent samples of \(m\) millilitres of the solution.
      The probability that at least 2 of these samples contain no particles is 0.1184
  2. Using the Statistical Tables provided, find the value of \(m\)