Question

For each of the following situations, calculate the expected value. a. Tanisha owns one share of IBM stock, which is currently trading at \(\$ 80 .\) There is a \(50 \%\) chance that the share price will rise to \(\$ 100\) and a \(50 \%\) chance that it will fall to \(\$ 70\). What is the expected value of the future share price? b. Sharon buys a ticket in a small lottery. There is a probability of 0.7 that she will win nothing, of 0.2 that she will win \(\$ 10,\) and of 0.1 that she will win \(\$ 50 .\) What is the expected value of Sharon's winnings? c. Aaron is a farmer whose rice crop depends on the weather. If the weather is favorable, he will make a profit of \(\$ 100\). If the weather is unfavorable, he will make a profit of \(-\$ 20\) (that is, he will lose money). The weather forecast reports that the probability of weather being favorable is 0.9 and the probability of weather being unfavorable is \(0.1 .\) What is the expected value of Aaron's profit?

Short answer

Answer: The expected values are as follows: a. IBM share price: $85 b. Sharon's lottery winnings: $7 c. Aaron's profit: $88

Step-by-step solution

a. Expected value of IBM share price

To calculate the expected value of the future IBM share price, we multiply the value of each outcome (share price) by its probability, and then sum the results. Hence, the expected value is: (Expected value) = (share price when rises * probability of rise) + (share price when falls * probability of fall) = \(( 100 * 0.50) + (70 * 0.50)\) = \(50 + 35\) = \(85\) Therefore, the expected value of the future IBM share price is \( \$ 85\).

b. Expected value of Sharon's winnings in the lottery

To calculate the expected value of Sharon's winnings in the lottery, we multiply the value of each outcome (winnings) by its probability, and then sum the results. Hence, the expected value is: (Expected value) = (winnings when nothing * probability of nothing) + (winnings when 10 dollars * probability of 10 dollars) + (winnings when 50 dollars * probability of 50 dollars) = \(( 0 * 0.7) + (10 * 0.2) + (50 * 0.1)\) = \(0 + 2 +5\) = \(7\) Therefore, the expected value of Sharon's winnings in the lottery is \( \$ 7.\)

c. Expected value of Aaron's profit

To calculate the expected value of Aaron's profit, we multiply the value of each outcome (profit) by its probability, and then sum the results. Hence, the expected value is: (Expected value) = (profit when weather is favorable * probability of favorable weather) + (profit when weather is unfavorable * probability of unfavorable weather) = \((100 * 0.9) + (-20 * 0.1)\) = \(90 - 2\) = \(88\) Therefore, the expected value of Aaron's profit is \( \$ 88.\).

Key concepts

Understanding Probability

Probability is a measure of the likelihood that a certain event will occur. In essence, it's a way to quantify uncertainty. When we say there is a 50% chance of rain tomorrow, we're saying there is an equal likelihood of rain as there is not rain.
Probabilities are expressed as numbers between 0 and 1. A probability of 0 means the event will not happen, while a probability of 1 means it will happen. In real-life situations, probabilities can be used to predict outcomes, helping in decision-making.

• A probability of 0.5 indicates an event is equally likely to happen or not happen.
• If an event has multiple potential outcomes, probabilities for all outcomes must add up to 1.
• Probability is essential in various fields, including finance, to assess risks and make informed decisions.

What Are Outcomes?

Outcomes are the possible results that can occur from an event or situation. Understanding the different outcomes of a scenario helps in analyzing the overall picture.
In probability, outcomes are usually expressed as different values a variable can take. For instance, in a dice roll, the outcomes are the numbers one through six that the die shows.

• In business or finance, outcomes could be profit, loss, or any financial result.
• Identifying outcomes helps in calculating the likelihood of each one, crucial for making decisions under uncertainty.
• In decision-making scenarios, like investments or lotteries, outlining potential outcomes is the first step in evaluating alternatives.

Calculation of Expected Value

The expected value is a fundamental concept in probability and statistics. It represents the average outcome if an experiment or process is repeated numerous times. Essentially, it combines all possible outcomes, weighted by their probabilities, to determine a single value.
Here’s how you can calculate the expected value:

• List all the possible outcomes.
• Assign probabilities to each outcome.
• Multiply each outcome by its probability.
• Sum all the products from the previous step.

The formula looks like this:\[E(X) = \sum \text{(Outcome value)} \times \text{(Probability of the outcome)}\]
The calculations you performed for IBM stock, Sharon's lottery, and Aaron's profit relied on this formula. Each scenario used different possible outcomes, like price changes or various winnings, to find an expected financial return.

Decision Making Under Uncertainty

In real-world scenarios, we often make decisions under uncertainty. This means we don't have all the information and cannot predict outcomes with certainty.
Expected value becomes a powerful tool in these situations. It helps assess the potential gains or losses of different strategies, providing a clearer view of long-term prospects.
Consider these points when making decisions under uncertainty:

• Identify all possible outcomes and their probabilities.
• Calculate the expected value to compare options.
• Evaluate the risks associated with each decision.
• Remember that a higher expected value doesn't guarantee success in any single instance.

For individuals like Tanisha, Sharon, and Aaron, understanding expected value provides a way to make calculated choices amidst uncertain conditions. It shifts the focus from short-term outcomes to overall long-term benefits.