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?

Understanding the Problem

An investment consultant provides us with two pieces of information about the stock price's behavior in the upcoming week:

1.     Odds against the price going up are 2:1. This means that for every 2 times the price does not go up, it is expected to go up once.

2.     Odds in favour of the price remaining the same are 1:3. This means that for every 1 time the price remains the same, it is expected to change in some other way (either go up or go down) 3 times.

With this information, we are asked to determine the probability that the stock price will go down in the next week.

Interpreting Odds and Probabilities

To solve this, we first need to understand the relationship between odds and probabilities.

• Odds of "A to B" means that the probability of event A occurring is $\frac{A}{A + B}$A+BA​. This is important because it helps us translate the given odds into probabilities that we can work with.

Step 1: Converting Odds Against the Price Going Up

The odds against the price going up are 2:1. This means that for every 2 situations where the price does not go up, there is 1 situation where the price does go up.

This implies the total number of possible outcomes is 3 (2 against and 1 for the price going up). Therefore, the probability that the price will go up is:

$$P(\text{Price Goes Up}) = \frac{1}{2 + 1} = \frac{1}{3}$$

Since the price can either go up, stay the same, or go down, the remaining probability must be distributed between the price staying the same and the price going down.

Step 2: Converting Odds in Favor of the Price Remaining the Same

The odds in favor of the price remaining the same are 1:3. This means that for every 1 time the price stays the same, there are 3 situations where the price changes (either goes up or goes down).

This implies the total number of possible outcomes is 4 (1 for staying the same and 3 for changing). Therefore, the probability that the price will remain the same is:

$$$$

$$P(\text{Price Remains the Same}) = \frac{1}{1 + 3} = \frac{1}{4}$$

Step 3: Determining the Probability of the Price Going Down

We know that the three possible outcomes for the stock price are:

1.     It goes up.

2.     It stays the same.

3.     It goes down.

The sum of the probabilities of these three outcomes must equal 1, since one of these outcomes must happen. Therefore, we have:

$$$$

$$P(\text{Price Goes Up}) + P(\text{Price Remains the Same}) + P(\text{Price Goes Down}) = 1$$

Substituting the probabilities we know:

$$$$

$$\frac{1}{3} + \frac{1}{4} + P(\text{Price Goes Down}) = 1$$

To solve for $P(\text{Price Goes Down})$P(Price Goes Down), we first need to add the fractions $\frac{1}{3}$31​ and $\frac{1}{4}$41​. To do this, we find a common denominator:

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

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

Thus:

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

$$\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}$$

Now we can substitute this into the equation:

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

$$\frac{7}{12} + P(\text{Price Goes Down}) = 1$$

Solving for $P(\text{Price Goes Down})$P(Price Goes Down):

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

$$P(\text{Price Goes Down}) = 1 - \frac{7}{12} = \frac{12}{12} - \frac{7}{12} = \frac{5}{12}$$

Conclusion

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

Final Note

While the solution is mathematically straightforward, the reasoning behind translating odds into probabilities and then solving for the unknown probability of the price going down is essential for making accurate predictions based on uncertain events like stock market movements. By understanding how odds and probabilities work, we can make more informed decisions about potential outcomes.

$$$$

This explanation should cover the full process in detail, from interpreting the given odds to solving the problem step by step.

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