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?

Introduction to Probability and Odds

To begin with, we need to understand the difference between odds and probability, as the question relies on both concepts. Probability is a mathematical measure of the likelihood that a certain event will occur, and it ranges from 0 (impossible event) to 1 (certain event). For example, if there’s a 50% chance of rain tomorrow, the probability is 0.5.

Odds, on the other hand, are a ratio of the likelihood of an event happening versus the likelihood of it not happening. Odds are often stated as "odds in favour" or "odds against" an event. The odds in favour of an event are the ratio of the number of favorable outcomes to the number of unfavorable outcomes. Conversely, the odds against an event are the ratio of the number of unfavorable outcomes to the number of favorable outcomes.

Problem Breakdown

In the given question, an investment consultant predicts that:

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

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

The task is to determine the probability that the price of the stock will go down during the next week.

To solve this problem, we need to clarify what each event entails:

• "The price goes up" is one possible outcome.
• "The price stays the same" is another possible outcome.
• "The price goes down" is the third possible outcome.

Together, these three outcomes (going up, staying the same, or going down) cover all possible scenarios for the price of the stock. This means that the sum of the probabilities for all three outcomes must equal 1.

Step 1: Understanding Odds and Converting to Probabilities

Odds Against the Price Going Up

The odds against the price of the stock going up are given as 2:1. This means that for every 3 possible outcomes, 2 of them are that the price will not go up, and 1 of them is that the price will go up.

To convert this into probability, we use the following formula:

$$P(\text{Up}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$$

In this case, the number of favorable outcomes (for the stock going up) is 1, and the total number of outcomes (up + not up) is 3 (since the odds are 2:1). Therefore, the probability of the price going up is:

$$$$

$$P(\text{Up}) = \frac{1}{3}$$

Odds In Favour of the Price Remaining the Same

The odds in favour of the price remaining the same are given as 1:3. This means that for every 4 possible outcomes, 1 of them is that the price will remain the same, and 3 of them are that the price will not remain the same (it either goes up or down).

To convert these odds into probability, we use a similar process. The number of favorable outcomes (for the price remaining the same) is 1, and the total number of outcomes (same + not same) is 4. Therefore, the probability of the price remaining the same is:

$$$$

$$P(\text{Same}) = \frac{1}{4}$$

Probability of the Price Going Down

The sum of the probabilities for all three outcomes must be equal to 1:

$$$$

$$P(\text{Up}) + P(\text{Same}) + P(\text{Down}) = 1$$

Substituting the probabilities for the "Up" and "Same" outcomes:

$$$$

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

To find the probability that the price goes down, we solve for $P(\text{Down})$P(Down):

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

$$P(\text{Down}) = 1 - \left( \frac{1}{3} + \frac{1}{4} \right)$$

We need to add $\frac{1}{3}$31​ and $\frac{1}{4}$41​. To do this, we need a common denominator:

$$$$

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

Now add the fractions:

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

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

Substitute this back into the equation:

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

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

Conclusion

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

Final Thoughts on the Calculation

This problem demonstrates how odds can be translated into probabilities. The key is recognizing that odds give us a ratio of favorable to unfavorable outcomes, and from this, we can derive the probability. The most important aspect of solving this problem was understanding that the sum of all probabilities must equal 1, which allowed us to solve for the unknown probability of the price going down.

Now that we've completed the calculation and arrived at the probability, let's briefly summarize the steps taken:

1.      We were given the odds for two possible outcomes: the price going up and the price staying the same.

2.      We converted those odds into probabilities.

3.      We used the fact that the total probability must add up to 1 to find the probability that the price would go down.

$$$$

This approach can be generalized to other problems involving odds and probability, and the same method can be applied to different events, provided that the sum of probabilities is always 1.

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