Problem 30 - Basic Statistics for Business and Economics with Student

Question

An investor owns three common stocks. Each stock, independent of the others, has equally likely chances of (1) increasing in value, (2) decreasing in value, or (3) remaining the same value. List the possible outcomes of this experiment. Estimate the probability at least two of the stocks increase in value.

Step-by-Step Solution

Answer

The probability that at least two stocks increase in value is approximately 0.259.

Step 1: Understand the Possible Outcomes for One Stock

Each stock has three possible outcomes: 1. It increases in value. 2. It decreases in value. 3. It remains the same. These outcomes are independent for each stock.

Step 2: Calculate the Total Number of Outcomes for Three Stocks

Since each of the three stocks can have one of three outcomes (increase, decrease, remain the same), we calculate the total different combinations using the formula for permutations: \[ 3 \times 3 \times 3 = 27 \]Thus, there are 27 possible outcomes for the entire portfolio of three stocks.

Step 3: List All Possible Outcomes

Each stock can independently be in one of the three states: Increase (I), Decrease (D), Remain (S). Therefore, list all combinations of these outcomes for three stocks: 1. III 2. IID 3. IIS 4. IDI 5. IDD 6. IDS 7. ISI 8. ISD 9. ISS 10. DII 11. DID 12. DIS 13. DDI 14. DDD 15. DDS 16. DSI 17. DSD 18. DSS 19. SII 20. SID 21. SIS 22. SDI 23. SDD 24. SDS 25. SSI 26. SSD 27. SSS

Step 4: Identify Favorable Outcomes

We need at least two stocks to increase in value, so examine the list for outcomes where this condition is met: 1. III 2. IID 3. IIS 4. IDI 5. ISI 6. DII 7. SII These are 7 favorable outcomes.

Step 5: Calculate the Probability of Favorable Outcomes

The probability of the desired event (at least two stocks increase) is calculated by dividing the number of favorable outcomes by the total number of outcomes:\[ \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Possible Outcomes}} = \frac{7}{27} \approx 0.259 \]

Key Concepts

Independent Events

One crucial idea in probability theory is the concept of independent events. In this context, independent events refer to occurrences where the result of one event does not influence the outcome of another. This means that each of the investor's stocks behaves independently when it comes to changes in their value.
Each stock can either increase, decrease, or stay the same value without being affected by the performance of the other two stocks. This independence allows us to approach the problem with more straightforward calculations.
In simpler terms, think of each stock as rolling its own die, where each face represents a different outcome. The result of one die (or stock) will not affect the roll of another, maintaining their independence.

Outcomes Calculation

Calculating the possible outcomes involves understanding each stock's potential changes. Given that each stock can have any of three conditions (increased, decreased, or unchanged), exploring the combination of these conditions is key.
For three stocks, each having three outcomes, the total number of possible combinations is calculated by multiplying the possible outcomes per stock:

• Outcome 1: Increases in value
• Outcome 2: Decreases in value
• Outcome 3: Remains the same

Thus, the total number of outcomes is \(3 \times 3 \times 3 = 27\).
This highlights how quickly possibilities increase in scenarios involving multiple independent events, showcasing the power of combinatorial calculations in probability.

Probability Estimation

Estimating the probability of a specific scenario, such as at least two stocks increasing in value, involves identifying the outcomes that fit this requirement.
From the list of 27 potential outcomes, you need to count how many meet your condition. In this example, 7 combinations involve at least two stocks increasing in value.
To estimate probability, divide the number of favorable outcomes by the total number of outcomes:

• Favorable Outcomes _= 7_
• Total Outcomes _= 27_

Thus, the calculation is \(\frac{7}{27} \approx 0.259\).
This result, approximately 25.9%, gives an investor a clearer understanding of the likelihood of a favorable scenario occurring, tying back to informed decision-making based on probabilistic analysis.