What would be a good problem Statement for the below problem.

What are the assumptions of the below problem

What are the formulas used to solve the problems

Rob is practicing his new Dart Game. He has a 20" diameter target, which has 4 regions consisting of Concentric circles. The inner circle has a diameter 5". the next 10" and the next 15". Rob only counts throws which hit the target. He gets 10 points for landing his dart in the outer ring, 20 for the next, 30 for the next and 100 points for the center. Assume that the probability of hitting a particular ring is proportional to the area of the ring. Find the Expected value.

Here is the solved problem.

The expected value is 22.5 points.

Explanation:

First we calculate the total area as:

Diameter of the outer circle = 20'', radius = 10''.

Area = * R => * (10) = 100 in

Area of circle that is worth 100 points:

Area = * (2.5) = 6.25 in

Area of circle that is worth 30 points will be the area of a circle with radius equal to (10/2 = 5) - the area of the inner circle.

Area = (5) = 25 - 6.25 = 18.75 in

Area of circle that is worth 20 points:

Area = *(7.5) = 56.25-25 =31.25 in

Area of circle that is worth 10 points:

Area = (10) - 56.25 = 43.75 in

Now we can find the % of each point:

% of hitting 100 points: 6.25/100 = 6.25%

% of hitting 30 points: 18.75/100 = 18.75%

% of hitting 20 points: 31.25/100 = 31.25%

% of hitting 10 points: 43.75/100 = 43.75%

Total = 6.25+18.75+31.25+43.75 = 100%

Now the expected value can be found as the summ of % * points.

100(6.25%)+30(18.75%)+20(31.25%)+10(43.75%) = 22.5 points