Analyze the following decision tree. After determining the missing probabilities, identify which alternative (Option 1 or Option 2) has the higher expected payoff. What is the expected payoff?
A) Option 1, with an expected payoff of less than $35
B) Option 1, with an expected payoff of no less than $35
C) Option 2, with an expected payoff of less than $35
D) Option 2, with an expected payoff of more than $35

Question

Analyze the following decision tree. After determining the missing probabilities, identify which alternative (Option 1 or Option 2) has the higher expected payoff. What is the expected payoff? 
A) Option 1, with an expected payoff of less than $35 
B) Option 1, with an expected payoff of no less than $35 
C) Option 2, with an expected payoff of less than $35 
D) Option 2, with an expected payoff of more than $35

Quizplus · Accepted Answer

To calculate the expected payoffs, we need to fill in the missing probabilities and then work our way backward.

Starting from the rightmost branch, we have:

- Option 2, Event A: Probability is not given, but we know that the payoff is $60.
- Option 2, Event B: Probability is 0.6, and the payoff is $40.
- Option 2, Event C: Probability is 0.4, and the payoff is $0.

Moving one step to the left, we have:

- Option 1, Event D: Probability is not given, but we know that the payoff is $50.
- Option 1, Event E: Probability is 0.75, and the payoff is $30.
- Option 1, Event F: Probability is 0.25, and the payoff is -$20.

Now we can solve for the missing probabilities by using the fact that the probabilities for any given event must add up to 1. Let x be the probability for Event A, then we have:

- x + 0.6 = 1 (from the probabilities of Events A and B)
- x = 0.4
- Option 2, Event A has a probability of 0.4.

Similarly, let y be the probability for Event D, then we have:

- y + 0.75 = 1 (from the probabilities of Events D and E)
- y = 0.25
- Option 1, Event D has a probability of 0.25.

Now we can calculate the expected payoffs for each option:

- Option 1: E(D) = 0.25($50) + 0.75($30) + 0.25(-$20) = $36.25
- Option 2: E(D) = 0.4($60) + 0.6($40) + 0.4($0) = $44

Therefore, Option 2 has the higher expected payoff, which is more than $35. The answer is D.

Analyze the Following Decision Tree