Problem 53 - Algebra for College Students

Question

A contractor estimates a probability of \(0.7\) of making \(\$ 20,000\) on a building project and a probability of \(0.3\) of losing \(\$ 10,000\) on the project. What is his mathematical expectation? \(\$ 11,000\)

Step-by-Step Solution

Answer

The contractor's mathematical expectation is $11,000.

Step 1: Define the Probabilities and Outcomes

Identify the probabilities and their corresponding outcomes given by the contractor. He has a probability of 0.7 to gain $20,000 and a probability of 0.3 to lose $10,000.

Step 2: Calculate Expected Value - Gain

Multiply the probability of gaining money by the amount gained: \(0.7 \times 20,000 = 14,000\).

Step 3: Calculate Expected Value - Loss

Multiply the probability of losing money by the amount lost: \(0.3 \times (-10,000) = -3,000\).

Step 4: Sum the Expected Values

Add the results from Step 2 and Step 3 together to find the mathematical expectation: \(14,000 + (-3,000) = 11,000\).

Key Concepts

Probability

Probability is a way to measure how likely an event is to happen. In this exercise, the contractor assesses two potential scenarios. One has a probability of 0.7, which indicates a 70% chance of earning $20,000. The other scenario holds a probability of 0.3, meaning there's a 30% chance of incurring a loss of $10,000. With probability, the total likelihood of all possible outcomes in a situation must add up to 1. Therefore, in this exercise, the chances of gain and loss sum up to 1, confirming that all possible outcomes have been considered. Understanding probability helps in predicting future events and making informed decisions based on likelihoods rather than just intuition.

Expected Value

Expected value is a fundamental concept in probability, offering a way to calculate an average outcome when dealing with various risky scenarios. It provides the expected long-term result of different elements of chance. In the exercise, the expected value is calculated by looking at both possible outcomes - a gain or a loss - and their probabilities. To find the expected value, calculate the probability-weighted average of all potential outcomes. This means multiplying each outcome by its respective probability and then adding these products. In this example, the expected value is computed by multiplying $20,000 (gain) by 0.7 and $-10,000 (loss) by 0.3, and summing these results to get $11,000.

Gain and Loss Calculation

When working with random variables in probabilities, calculating the potential gains and losses is crucial. In this scenario, the contractor can either gain $20,000 or lose $10,000. The calculation involves:

• Multiplying the probability of a gain by the amount of the gain, resulting in $14,000 (i.e., 0.7 multiplied by $20,000).
• Multiplying the probability of a loss by the amount of the loss, giving a negative value of $3,000 (i.e., 0.3 multiplied by $-10,000).

These calculations help to determine the expected impact on cash flow, allowing an understanding of financial risks and rewards.

Probability Outcomes

Probability outcomes are the possible results of a chance-based situation, each paired with a specific probability. In this exercise, the outcomes are a gain of $20,000 and a loss of $10,000. It's essential to clearly define each potential outcome and its probability to accurately assess the risk and expected returns. It ensures that each outcome's influence on the overall expectation is fully considered. By analyzing probability outcomes, individuals and businesses can strategize effectively by gauging potential financial impact and adjusting plans to accommodate both favorable and unfavorable possibilities.