Question

An investor has the option of investing in three of five recommended stocks. Unknown to her, only two will show a substantial profit within the next 5 years. If she selects the three stocks at random (giving every combination of three stocks an equal chance of selection), what is the probability that she selects the two profitable stocks? What is the probability that she selects only one of the two profitable stocks?

Short answer

Answer: The probability of selecting both profitable stocks is 30% (3/10) and selecting only one profitable stock is 60% (3/5).

Step-by-step solution

Calculate the total number of possible combinations

The investor wants to select 3 stocks out of 5. We will use the combination formula to find the total number of possible ways to do this: \(nCr = \frac{n!}{r!(n-r)!}\) where n is the total number of elements and r is the number of elements to be chosen. Here, n = 5 and r = 3. \(C(5,3) = \frac{5!}{3!(5-3)!} = \frac{5!}{3!2!} = \frac{120}{12} = 10\) So, there are 10 possible ways to choose 3 stocks out of 5.

Calculate the probability of selecting both profitable stocks

The investor needs to select both profitable stocks, so the task is choosing 2 out of 2 profitable stocks and 1 out of 3 non-profitable stocks. The number of ways to choose 2 profitable stocks out of 2 is C(2,2) = 1 (since there are only two profitable stocks and both need to be selected). The number of ways to choose 1 non-profitable stock out of 3 is C(3,1) = 3. Total successful outcomes: 1 x 3 = 3. So, probability = successful outcomes / total outcomes = \(\frac{3}{10}\) .

Calculate the probability of selecting only one profitable stock

Now, the investor needs to select only one profitable stock, i.e., choosing 1 out of 2 profitable stocks and 2 out of 3 non-profitable stocks. The number of ways to choose 1 profitable stock out of 2 is C(2,1) = 2. The number of ways to choose 2 non-profitable stocks out of 3 is C(3,2) = 3. Total successful outcomes: 2 x 3 = 6. So, probability = successful outcomes / total outcomes = \(\frac{6}{10} = \frac{3}{5}\). The probability that the investor selects both profitable stocks is \(\frac{3}{10}\) (30%) and the probability that she selects only one profitable stock is \(\frac{3}{5}\) (60%).

Key concepts

Combinations

Combinations play a crucial role in understanding probability problems like the one in our exercise. When we talk about combinations, we refer to the way we can select items from a larger group without considering the order. Simply put, it’s about grouping things together.

To find combinations, we use the formula:
\[ C(n,r) = \frac{n!}{r!(n-r)!} \]
where **n** is the total number of items, **r** is the number of items we want to choose, and "!" denotes factorial, which means multiplying a series of descending natural numbers. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.

In the exercise, the investor wants to pick 3 stocks out of a possible 5, leading us to calculate
\[ C(5,3) = \frac{5!}{3!(5-3)!} = \frac{120}{12} = 10 \].
This means there are 10 different ways to choose 3 stocks from 5. Understanding the concept of combinations allows us to explore various outcomes and probabilities, simplifying complex investment decisions among limited choices.

Profitability

Profitability is a key metric used to determine the success of any investment. It refers to the extent to which an investment generates a significant financial return over time.

In the context of our problem, understanding profitability involves assessing the chances of selecting profitable stocks randomly. Out of the 5 stocks available to the investor, only 2 are profit-generating. Thus, determining the likelihood of selecting one or both profitable stocks is a matter of calculating favorable outcomes.

Let's break this down:

• Selecting Both Profitable Stocks: The probability for this is calculated as the number of favorable selections (3 combinations that include both profitable stocks) out of total possible stock selections (10 combinations), giving us \( \frac{3}{10} \) or 30% probability.
• Selecting Only One Profitable Stock: Here, our favorable outcomes (selecting one profitable and two non-profitable stocks) are 6, leading to a probability of \( \frac{6}{10} = \frac{3}{5} \) or 60%.

Grasping how profitability impacts investment decisions helps investors assess their chances realistically.

Investment strategy

Investment strategy is about making informed decisions to optimize returns and manage risks effectively. For investors, strategy involves analyzing probabilities and potential outcomes to choose the best course of action.

The given problem emphasizes the importance of selecting stocks based on probability. Although the selection is random, having prior knowledge of profitable stocks could improve decision-making.

Key elements to consider in an investment strategy include:

• Understanding Combinations: Knowing how to calculate combinations helps estimate the number of ways to achieve desired investment goals.
• Assessing Profitability: Evaluating the likelihood of choosing profitable investments ensures more informed choices.
• Risk Management: By determining probabilities, investors can balance risk and reward, optimizing their portfolio selections.

Overall, incorporating probability theory into an investment strategy enables investors to manage their portfolios smarter, paving the way for successful financial growth.