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# Probability

Assume that the dollar loss, L, associated with being robbed by a mugger on the street is \$3, and that being robbed occurs with a probability of p. Suppose an individual can influence p by exercising caution, but doing so will be costly. Let the cost, C, required to achieve probability p be given by C=4(1-p)^3. Assume that the individual is risk neutral.

a. If insurance is not available, what probability of loss will the agent choose?

Now, assume an insurance policy that pays the full amount of the loss (if it occurs) is now available at a total premium or price Z (i.e., the policy cost is Z, not ZL). Then,

b. If insurance has already been purchased, what probability of a loss will
insurance company charge in order to "break even?"
c. Using the answers obtained from part a and b, prove that a risk neutral
agent will not purchase any insurance at the premium that must be
charged in order for the insurance company to break even.

#### Solution Preview

Part a)
If the individual is risk neutral. He will chose a probability at which his expected value from the cost of reducing the probability is zero.
Thus,
In case of loss, his payoff will -C and in case when there is no loss his payoff will be +3-C.
The expected payoff will be
-C*p + (3-C)*(1-p)=0
-4(1-p)^3*p + [3-4(1-p)^3*(1-p)=0
Solving you will get the answer for p.
The value for p will be 0.1339

Part b)
If the insurance is already been taken, the agent will be insured ...

#### Solution Summary

Probability is assessed.

\$2.19