Finding minimum n for P(X≥k) threshold

Questions requiring the smallest n such that the probability of at least k occurrences satisfies a condition (e.g., P(X≥1) = 0.9, or P(X≥n) < 0.05).

5 questions · Standard +0.2

5.02i Poisson distribution: random events model5.02k Calculate Poisson probabilities
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CAIE S2 2008 June Q6
8 marks Standard +0.3
6 People arrive randomly and independently at the elevator in a block of flats at an average rate of 4 people every 5 minutes.
  1. Find the probability that exactly two people arrive in a 1-minute period.
  2. Find the probability that nobody arrives in a 15 -second period.
  3. The probability that at least one person arrives in the next \(t\) minutes is 0.9 . Find the value of \(t\).
CAIE S2 2019 March Q5
8 marks Standard +0.8
5 The number of eagles seen per hour in a certain location has the distribution \(\operatorname { Po } ( 1.8 )\). The number of vultures seen per hour in the same location has the independent distribution \(\operatorname { Po } ( 2.6 )\).
  1. Find the probability that, in a randomly chosen hour, at least 2 eagles are seen.
  2. Find the probability that, in a randomly chosen half-hour period, the total number of eagles and vultures seen is less than 5 .
    Alex wants to be at least \(99 \%\) certain of seeing at least 1 eagle.
  3. Find the minimum time for which she should watch for eagles.
OCR S2 2006 June Q6
14 marks Moderate -0.3
6 Customers arrive at a post office at a constant average rate of 0.4 per minute.
  1. State an assumption needed to model the number of customers arriving in a given time interval by a Poisson distribution. Assuming that the use of a Poisson distribution is justified,
  2. find the probability that more than 2 customers arrive in a randomly chosen 1 -minute interval,
  3. use a suitable approximation to calculate the probability that more than 55 customers arrive in a given two-hour interval,
  4. calculate the smallest time for which the probability that no customers arrive in that time is less than 0.02 , giving your answer to the nearest second.
OCR MEI S2 2009 June Q2
19 marks Moderate -0.3
2 Jess is watching a shower of meteors (shooting stars). During the shower, she sees meteors at an average rate of 1.3 per minute.
  1. State conditions required for a Poisson distribution to be a suitable model for the number of meteors which Jess sees during a randomly selected minute. You may assume that these conditions are satisfied.
  2. Find the probability that, during one minute, Jess sees
    (A) exactly one meteor,
    (B) at least 4 meteors.
  3. Find the probability that, in a period of 10 minutes, Jess sees exactly 10 meteors.
  4. Use a suitable approximating distribution to find the probability that Jess sees a total of at least 100 meteors during a period of one hour.
  5. Jess watches the shower for \(t\) minutes. She wishes to be at least \(99 \%\) certain that she will see one or more meteors. Find the smallest possible integer value of \(t\).
OCR S2 2010 January Q9
16 marks Standard +0.3
Buttercups in a meadow are distributed independently of one another and at a constant average incidence of 3 buttercups per square metre.
  1. Find the probability that in 1 square metre there are more than 7 buttercups. [2]
  2. Find the probability that in 4 square metres there are either 13 or 14 buttercups. [3]
  3. Use a suitable approximation to find the probability that there are no more than 69 buttercups in 20 square metres. [5]
    1. Without using an approximation, find an expression for the probability that in \(m\) square metres there are at least 2 buttercups. [2]
    2. It is given that the probability that there are at least 2 buttercups in \(m\) square metres is 0.9. Using your answer to part (a), show numerically that \(m\) lies between 1.29 and 1.3. [4]