Calculate single probability using Poisson approximation

1 On average, 2 people in every 10000 in the UK have a particular gene. A random sample of 6000 people in the UK is chosen. The random variable \(X\) denotes the number of people in the sample who have the gene. Use an approximating distribution to calculate the probability that there will be more than 2 people in the sample who have the gene.

1 A random variable \(X\) has the distribution \(\mathrm { B } \left( 4500000 , \frac { 1 } { 1000000 } \right)\).
Use a Poisson distribution to calculate an estimate of \(\mathrm { P } ( X \geqslant 4 )\). \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-03_2716_29_107_22}

1 On average 1 in 25000 people have a rare blood condition. Use a suitable approximating distribution to find the probability that fewer than 2 people in a random sample of 100000 have the condition.

2 It is given that on average one car in forty is yellow. Using a suitable approximation, find the probability that, in a random sample of 130 cars, exactly 4 are yellow.

5 A receptionist receives incoming telephone calls and should connect them to the appropriate department. The probability of them being connected to the wrong department on the first attempt is 0.05 A random sample of 8 calls is taken.

1. Find the probability that at least 2 of these calls are connected to the wrong department on the first attempt. The receptionist receives 1000 calls each day.
2. Use a Poisson approximation to find the probability that exactly 45 callers are connected to the wrong department on the first attempt in a day. The total time, \(T\) seconds, taken for a call to be answered by a department has a continuous uniform distribution over the interval [10,50]
3. Find \(\mathrm { P } ( T > 16 )\) The number of calls the receptionist receives in a one-minute interval is modelled by a Poisson distribution with mean 6 The receptionist receives a call from Jia and tries to connect it to the right department.
4. Find the probability that in the next 40 seconds Jia's call is answered by the right department on the first attempt and the receptionist has received no other calls.

4. The random variable \(X \sim \mathrm {~B} ( 150,0.02 )\). Use a suitable approximation to estimate \(\mathrm { P } ( X > 7 )\).

A bag contains a large number of counters of which \(15 \%\) are coloured red. A random sample of 30 counters is selected and the number of red counters is recorded.

1. Find the probability of no more than 6 red counters in this sample. A second random sample of 30 counters is selected and the number of red counters is recorded.
2. Using a Poisson approximation, estimate the probability that the total number of red counters in the combined sample of size 60 is less than 13.
   

An administrator makes errors in her typing randomly at a rate of 3 errors every 1000 words.

1. In a document of 2000 words find the probability that the administrator makes 4 or more errors. The administrator is given an 8000 word report to type and she is told that the report will only be accepted if there are 20 or fewer errors.
2. Use a suitable approximation to calculate the probability that the report is accepted.

1.5% of the population of the UK can be classified as 'very tall'.

1. The random variable \(X\) denotes the number of people in a sample of \(n\) people who are classified as very tall. Given that E\((X) = 2.55\), find \(n\). [2]
2. By using the Poisson distribution as an approximation to a binomial distribution, calculate an approximate value for the probability that a sample of size 210 will contain fewer than 3 people who are classified as very tall. [3]

The random variable \(Y\) has the distribution B(140, 0.03). Use a suitable approximation to find P(\(Y = 5\)). Justify your approximation. [5]