Question

A risk-averse individual is offered a choice between a gamble that pays \(\$ 1000\) with a probability of \(25 \%\) and \(\$ 100\) with a probability of \(75 \%,\) or a payment of \(\$ 325 .\) Which would he choose?

Short answer

He would choose the certain payment of \(\$325\).

Step-by-step solution

Define the Problem

Our task is to determine whether a risk-averse individual would prefer a gamble with two possible outcomes or a certain payment. The gamble offers a chance to win \(\\(1000\) with a probability of 25% and \(\\)100\) with a probability of 75%, while the certain payment is \(\$325\).

Calculate the Expected Value of the Gamble

The expected value (EV) of a gamble is calculated by multiplying each outcome by its probability and summing the results. Here, the EV is calculated as follows: \[ \text{EV} = (1000 \times 0.25) + (100 \times 0.75) = 250 + 75 = 325 \]

Compare the Expected Value and Certain Payment

The expected value of the gamble is \(\$325\), which is exactly the same as the certain payment offered. A risk-averse individual typically prefers certainty over a gamble, especially when faced with equal expected values.

Key concepts

Expected Value

The concept of expected value plays a crucial role in decision-making scenarios, particularly when evaluating different choices that involve uncertainty. Simply put, the expected value is the average amount one stands to gain or lose per gamble if the gamble is repeated many times. For those interested in financial aspects, understanding expected value can aid in making sound investment decisions.

To calculate the expected value, you multiply the outcome of each possibility by the probability of that outcome occurring and then sum these products. In our exercise, we have two potential outcomes: winning $1000 with a probability of 25% and winning $100 with a probability of 75%. We compute the expected value using the formula:

• First outcome: $1000 $ imes$ 0.25 = $250
• Second outcome: $100 $ imes$ 0.75 = $75

Adding these results gives us $325, indicating the expected value of the gamble.

Probability

Probability measures how likely an event is to occur and is expressed as a number between 0 and 1. A probability of 0 means an event will not happen, while a probability of 1 means it will definitely happen.

In the context of the exercise, probabilities are assigned to each of the possible outcomes of the gamble. We know:

• There's a 25% chance (0.25 probability) of winning $1000.
• There's a 75% chance (0.75 probability) of winning $100.

These numbers represent the likelihood of each outcome, and they sum to 1, as all probabilities should. Knowing how to calculate and interpret probabilities can help us understand the risk and potential rewards associated with different decisions.

Decision Making Under Uncertainty

Decision making under uncertainty involves evaluating options where outcomes are not guaranteed, often requiring the consideration of probabilities and expected values. Individuals must decide whether to take risks or opt for safer, guaranteed outcomes.

In the exercise, the decision is between taking a gamble with equal expected value as a certain payment or simply accepting the secure payment. For a risk-averse person, who prefers to minimize uncertainty, the certain payment of $325 is attractive because it eliminates the variability associated with the gamble.

Typically, risk-averse individuals value stability and will often choose guaranteed outcomes even if the potential for higher returns exists in uncertain scenarios. By understanding individual risk preferences and the certainty-equivalent, better decisions can be made when facing uncertainty. This mindset helps in navigating financial decisions, insurance considerations, and investment strategies.