$1.$ Suppose that Kate and Anne enter into a pooling arrangement for auto losses. Assume that both women have the following loss distributions and that losses are independent.

Loss$=$$$50,000$ with $0\mathrm{.}02\%$ probability

$$0$ with a $0\mathrm{.}98$ percent probability

a$.)$ a$)$ Write out the possible outcomes and the probability of each outcome for Kate and Anne after they enter into a pooling arrangement. That is$,$ write out the probability distribution for each of the women after they enter into a pooling arrangement.

b$.)$ b$)$ Calculate the expected loss to each person prior to and subsequent to entering into a pooling arrangement.

c$.)$ c$)$ Calculate the standard deviation of the loss distribution to each person prior to and subsequent to entering into a pooling arrangement. What happens to the standard deviation subsequent to the pooling arrangement?

d$.)$ d$)$ What will happen to the standard deviation if we increase the pool size to $100$ million?