N2012 P2 Q10

Gold coins are found scattered throughout an archaeological site.

1. State two conditions needed for the number of gold coins found in a randomly chosen region of area 1 square metre to be well modelled by a Poisson distribution.[2]

Assume that the number of gold coins in 1 square metre has the distribution \(\mathrm{Po}(0.8)\).

2. Find the probability that in 1 square metre there are at least 3 gold coins.[1]
3. It is given that the probability that 1 gold coin is found in \(x\) square metres is \(0.2\). Write down an equation for \(x\), and solve it numerically given that \(x<1\).[2]
4. Use a suitable approximation to find the probability that in 100 square metres there are at least 90 gold coins. State the parameter(s) of the distribution that you use.[3]

Pottery shards are also found scattered throughout the site. The number of pottery shards in 1 square metre is an independent random variable with the distribution \(\mathrm{Po}(3)\). Use suitable approximations, whose parameters should be stated, to find

5. the probability that in 50 square metres the total number of gold coins and pottery shards is at least 200,[4]
6. the probability that in 50 square metres there are at least 3 times as many pottery shards as gold coins.[3]