Question

Now suppose each urn has a mixture of red and pink balls. Urn θ1 has an average of 4 red and 6 pink balls, and θ2 has an average of 9 red balls and 1 pink ball. There are 800 urns of type θ1 and 200 urns of type θ2 for a total of 1000 urns. All balls from all urns are taken out and mixed in a big bucket. You are to draw a ball at random (with eyes closed). The person hosting the game could identify if the ball came from urn θ1 or urn θ2. Suppose you picked a red ball. Based on the information, you are to develop a decision tree and determine which contract is the best alternative. The following steps are suggested: 2a) Calculate the total number of red balls in the 800θ1 and 200θ2 urns. 2b) Calculate the total number of red balls in all 1000 urns. 2c) Calculate the probability of a single ball observed to be red when removed at random from a θ1 urn and the probability of a single ball observed to be red when removed at random from a θ2 urn. 2d) Incorporate the contract and probability information based only on the single observation of a randomly removed red ball into a decision tree and calculate the expected monetary value for each contract alternative. 2e) Determine the optimum decision based on the maximum expected monetary value.

Answer 1

Choose contract θ1 as it has the maximum expected monetary value.

Step-by-step explanation:

In order to determine the best contract, we need to calculate the expected monetary value for each alternative based on the probability of drawing a red ball from each type of urn. First, calculate the total number of red balls in the 800θ1 urns and 200θ2 urns. Then, find the total number of red balls in all 1000 urns. Next, calculate the probability of drawing a red ball from a θ1 urn and a θ2 urn.

Incorporate this information into a decision tree and calculate the expected monetary value for each contract alternative based on a single observation of a randomly removed red ball.

The expected monetary value is calculated by multiplying the probability of drawing a red ball from a specific type of urn by the payoff associated with that urn, summing these values for each type of urn, and then selecting the contract with the maximum expected monetary value. In this scenario, contract θ1 is chosen as it yields the highest expected monetary value, making it the optimal decision.