Finding minimum n for P(X≥k) threshold

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 \(t\).

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.

1. During morning hours, employees arrive randomly at an office drinks dispenser at a rate of 2 every 10 minutes.

The number of employees arriving at the drinks dispenser is assumed to follow a Poisson distribution.

1. Find the probability that fewer than 5 employees arrive at the drinks dispenser during a 10-minute period one morning. During a 30 -minute period one morning, the probability that \(n\) employees arrive at the drinks dispenser is the same as the probability that \(n + 1\) employees arrive at the drinks dispenser.
2. Find the value of \(n\) During a 45-minute period one morning, the probability that between \(c\) and 12, inclusive, employees arrive at the drinks dispenser is 0.8546
3. Find the value of \(C\)
4. Find the probability that exactly 2 employees arrive at the drinks dispenser in exactly 4 of the 6 non-overlapping 10-minute intervals between 10 am and 11am one morning.

4. A centre for receiving calls for the emergency services gets an average of \(3 \cdot 5\) emergency calls every minute. Assuming that the number of calls per minute follows a Poisson distribution,

1. find the probability that more than 6 calls arrive in any particular minute. Each operator takes a mean time of 2 minutes to deal with each call, and therefore seven operators are necessary to cope with the average demand.
2. Find how many operators are required for there to be a \(99 \%\) probability that a call can be dealt with immediately. It is found from experience that a major disaster creates a surge of emergency calls. Taking the null hypothesis \(\mathrm { H } _ { 0 }\) that there is no disaster,
3. find the number of calls that need to be received in one minute to disprove \(\mathrm { H } _ { 0 }\) at the \(0.1 \%\) significance level.

5 Biological cell membranes have receptor molecules which perform various functions. It is known that the number of receptor molecules of a particular type can be modelled by a Poisson distribution with mean 6 per area of 1 square unit.

1. 1. Determine the probability that there are at least 10 of these receptor molecules in an area of 1 square unit.
   2. Determine the probability that there are fewer than 50 of these receptor molecules in an area of 10 square units.
2. A scientist is looking at areas of 1 square unit of cell membrane in order to find one which has at least 10 receptor molecules. Find the probability that she has to look at more than 20 to find such an area. It is known that the number of receptor molecules of another type in an area of 1 square unit can be modelled by the random variable \(X\) which has a Poisson distribution with mean \(\mu\). It is given that \(\mathrm { E } \left( X ^ { 2 } \right) = 12\).
3. Determine \(\mathrm { P } ( X < 5 )\).