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Decision Trees Young screenwriter Carl Draper has just finished his first script. It has action, drama, and humor, and he thinks it will be a blockbuster. He takes the script to every motion picture studio in town and tries to sell it but to no avail. Finally, ACME studios offers to buy the script for either (a) \(\$ 12,000\) or (b) 1 percent of the movie's profits. There are two decisions the studio will have to make. First is to decide if the script is good or bad, and second if the movie is good or bad. First, there is a 90 percent chance that the script is bad. If it is bad, the studio does nothing more and throws the script out. If the script is good, they will shoot the movie. After the movie is shot, the studio will review it, and there is a 70 percent chance that the movie is bad. If the movie is bad, the movie will not be promoted and will not turn a profit. If the movie is good, the studio will promote heavily; the average profit for this type of movie is \(\$ 20\) million. Carl rejects the \(\$ 12,000\) and says he wants the 1 percent of profits. Was this a good decision by Carl?

Step by step solution

01

Calculate the expected value if the script is bad

If the script is bad, there is a 90% chance that the studio will do nothing and throw the script out. This means Carl will earn no money. So, the expected value for Carl in this case is: \(0.9 \times \$0 = \$0\).

02

Calculate the expected value if the script is good and the movie is bad

If the script is good (10% chance) and the movie turns out bad (70% chance), then there is a 7% chance (0.1 x 0.7) the movie will not turn a profit. In this case, Carl's earnings will also be $0. So, the expected value for Carl in this case is: \(0.07 \times \$0 = \$0\).

03

Calculate the expected value if the script is good and the movie is good

If the script is good (10% chance) and the movie turns out good (30% chance), then there is a 3% chance (0.1 x 0.3) the movie will be promoted heavily and have an average profit of $20 million. As Carl has chosen 1% of profits, he will earn: \(\$20,\!000,\!000 \times 0.01 = \$200,\!000\). The expected value for Carl in this case is: \(0.03 \times \$200,\!000 = \$6,\!000\).

04

Calculate the total expected value for Carl

Now we can sum up the expected values from the three scenarios mentioned above: Total Expected Value = \((Expected Value: Script Bad) + (Expected Value: Script Good, Movie Bad) + (Expected Value: Script Good, Movie Good)\) Total Expected Value = \(\$0 + \$0 + \$6,000 = \$6,000\)

05

Compare the expected value with the flat payment offer

Finally, we can compare the calculated total expected value with the offer of \(\$12,000\): - Expected Value of choosing 1% of profits: \(\$6,000\) - ACME Studios' offer: \(\$12,000\) Since the expected value of 1% of profits is less than the \(\$12,000\) offer, we can conclude that Carl's decision to reject the \(\$12,000\) for 1% of profits was not a good decision.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expected Value

Understanding expected value is crucial in decision-making processes, especially in financial and risk-based evaluations. In simple terms, the expected value helps to determine the average outcome of a probability scenario.
It is calculated by multiplying each possible outcome by the likelihood of it occurring, and then adding all those values together.

• If an event can have multiple outcomes with different probabilities, the expected value provides a single number representing the overall average.
• This value is useful because it can compare different choices to find the one with the best average long-term results.

The concept is particularly relevant in Carl Draper's case. Carl faced a decision involving two offers from ACME studios: a certain $12,000 or 1% of potential movie profits. By calculating the expected value of potential movie profits, Carl could investigate if his decision was financially sound. Unfortunately, the expected value of his profit share amounted to $6,000, which was lower than the sure $12,000 offer.
This mathematical insight, via expected value, highlights that choosing the immediate payment would have been wiser in this instance.

Probability Analysis

Probability analysis involves evaluating the likelihoods of various outcomes and using them to forecast results. This analysis is essential when faced with uncertain situations. Each component of an event is given a probability that signifies how likely it is to occur, typically expressed as a percentage or a fraction.
In Carl's scenario, several probabilities were involved:

• A 90% chance that the script is bad, resulting in no profit.
• A 10% chance that the script is good.
• If good, a 70% chance the movie is bad, and a 30% chance the movie is good and profitable.

Analyzing these probabilities allows decision-makers to understand and prepare for the potential outcomes. It gives a numerical perspective to what might otherwise be gut-feeling decisions.
Probability analysis, in Carl's decision, revealed that the chance of earning significant profits was relatively low. This analytical process thus supported the conclusion that taking the fixed offer might have been the safer bet.

Risk Assessment

Risk assessment is the method used to identify and evaluate the financial risks involved in a particular decision. For Carl Draper, choosing the 1% profit option involved a deep understanding of potential hazards.
Risk assessment examines both positive and negative results:

• The high risk meant there was only a slim chance (3%) that he would gain substantial profit.
• A great risk assessment also considers the impact of negative scenarios, such as earning nothing if the movie fails.

This assessment demonstrated that while the reward could be high ($200,000 potential), the likelihood was low, and the downside was substantial compared to the guaranteed $12,000. Understanding this risk reassessment over two potential earnings shed light on why Carl's decision was risky. From an objective risk standpoint, minimizing downside potential would have likely suggested taking the upfront payment.

Decision Making

Decision making, especially in business and finance, is the process of choosing the best course of action among multiple alternatives. This requires evaluating expected value, probability, and risk.

• Decision-makers must consider not only the potential benefits but also the probabilities and risks of each outcome.
• Another essential part is comparison, where each option is weighed against others in terms of potential gain and likelihood.
• Creating a decision tree can help visualize these options and see at a glance which might offer the best trade-off of risk and reward.

In Carl's scenario, he had to choose between a safe $12,000 and the uncertainty of a profit share. Despite the allure of greater earnings, a thorough decision-making process based on expected value and probability analysis showed the fixed offer was more rational. Decision making in such contexts necessitates foregoing potential to secure a guarantee, indicative of risk aversion practices common among seasoned investors and business professionals.