An investment consultant predicts that the odds against the price of a certain stock will go up during the next week are 2:1 and the odds in favour of the price remaining the same are 1:3. What is the probability that the price of the stock will go down during the next week?

 Q. An investment consultant predicts that the odds against the price of a certain stock will go up during the next week are 2:1 and the odds in favour of the price remaining the same are 1:3. What is the probability that the price of the stock will go down during the next week?

Problem Statement

An investment consultant predicts that:

• The odds against the price of a certain stock going up during the next week are 2:1.
• The odds in favor of the price remaining the same are 1:3.

You are tasked with finding the probability that the price of the stock will go down during the next week.

Explanation and Solution

Let’s break the problem down step by step.

Step 1: Understanding Odds and Probabilities

Before solving the problem, we need to understand how odds relate to probabilities. The odds given in the problem are in the form of "odds against" or "odds in favor."

1.     Odds Against an Event: If the odds against an event are "A:B," it means that for every A failures, there are B successes. The probability $P(\text{event})$P(event) can be calculated using the formula:

$$P(\text{event}) = \frac{B}{A + B}$$

where A and B are the respective numbers of failures and successes.

2.     Odds in Favor of an Event: If the odds in favor of an event are "A:B," it means that for every A successes, there are B failures. The probability $P(\text{event})$P(event) is then:

$$$$

$$P(\text{event}) = \frac{A}{A + B}$$

This formula is similar but in reverse order.

Step 2: Translating Given Information

1.   The odds against the price going up are 2:1.

o   This means that for every 2 situations where the price does not go up, there is 1 situation where the price goes up. So, the probability of the price going up is:

$$$$

$$P(\text{price goes up}) = \frac{1}{2 + 1} = \frac{1}{3}$$

2.   The odds in favor of the price remaining the same are 1:3.

o   This means that for every 1 situation where the price remains the same, there are 3 situations where the price does not remain the same. The probability of the price remaining the same is:

$$$$

$$P(\text{price remains the same}) = \frac{1}{1 + 3} = \frac{1}{4}$$

Step 3: Determining the Remaining Probability

Since the stock price can either go up, remain the same, or go down, the total probability of these three outcomes must sum to 1. Therefore, the probability that the price will go down can be found by subtracting the probabilities of the other two outcomes (price going up and price remaining the same) from 1.

• Let’s denote the probability of the stock price going down as $P(\text{price goes down})$P(price goes down).

The total probability is:

$$$$

$$P(\text{price goes up}) + P(\text{price remains the same}) + P(\text{price goes down}) = 1$$

Substitute the known values:

$$$$

$$\frac{1}{3} + \frac{1}{4} + P(\text{price goes down}) = 1$$

Step 4: Solving for the Probability of the Price Going Down

To find $P(\text{price goes down})$P(price goes down), we first need to find a common denominator for the fractions $\frac{1}{3}$31​ and $\frac{1}{4}$41​. The least common denominator of 3 and 4 is 12.

$$\frac{\mathrm{}}{}$$

$$\frac{1}{3} = \frac{4}{12}, \quad \frac{1}{4} = \frac{3}{12}$$

Now, substitute these into the equation:

$$\frac{\mathrm{}}{}$$

$$\frac{4}{12} + \frac{3}{12} + P(\text{price goes down}) = 1$$

Simplify the left-hand side:

$$\frac{\mathrm{}}{}$$

$$\frac{7}{12} + P(\text{price goes down}) = 1$$

Solve for $P(\text{price goes down})$P(price goes down):

$$\frac{\mathrm{}}{}$$

$$P(\text{price goes down}) = 1 - \frac{7}{12} = \frac{12}{12} - \frac{7}{12} = \frac{5}{12}$$

Final Answer

The probability that the price of the stock will go down during the next week is $\frac{5}{12}$125​.

Conclusion

$$$$

In summary, the problem provided us with the odds of various outcomes related to the stock price. By translating these odds into probabilities and ensuring the total probability of all possible outcomes (price going up, remaining the same, or going down) sums to 1, we were able to determine that the probability of the stock price going down is $\frac{5}{12}$125​. This solution demonstrates how odds can be converted into probabilities and how basic probability rules can be applied to solve real-world problems.

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