Gold coins are found scattered throughout an archaeological site.
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 \(\mathrm{Po}(0.8)\).
Find the probability that in 1 square metre there are at least 3 gold coins.[1]
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]
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
the probability that in 50 square metres the total number of gold coins and pottery shards is at least 200,[4]
the probability that in 50 square metres there are at least 3 times as many pottery shards as gold coins.[3]