Scaled Poisson over time period

3 The number of e-readers sold in a 10-day period in a shop is modelled by the distribution \(\operatorname { Po } ( 5.1 )\). Use an approximating distribution to find the probability that fewer than 140 e-readers are sold in a 300-day period.

5 In a mine, a deposit of the substance pitchblende emits radioactive particles. The number of particles emitted has a Poisson distribution with mean 70 particles per second. The warning level is reached if the total number of particles emitted in one minute is more than 4350.

1. A one-minute period is chosen at random. Use a suitable approximation to show that the probability that the warning level is reached during this period is 0.01 , correct to 2 decimal places. You should calculate the answer correct to 4 decimal places.
2. Use a suitable approximation to find the probability that in 30 one-minute periods the warning level is reached on at least 4 occasions. (You should use the given rounded value of 0.01 from part (i) in your calculation.)

2 The number of neutrinos that pass through a certain region in one second is a random variable with the distribution \(\operatorname { Po } \left( 5 \times 10 ^ { 4 } \right)\). Use a suitable approximation to calculate the probability that the number of neutrinos passing through the region in 40 seconds is less than \(1.999 \times 10 ^ { 6 }\).

A botanist suggests that the number of a particular variety of weed growing in a meadow can be modelled by a Poisson distribution.

1. Write down two conditions that must apply for this model to be applicable. [2]

Assuming this model and a mean of 0.7 weeds per m², find

1. the probability that in a randomly chosen plot of size 4 m² there will be fewer than 3 of these weeds, [4]
2. Using a suitable approximation, find the probability that in a plot of 100 m² there will be more than 66 of these weeds. [6]