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?

The problem provided presents a typical situation involving odds and probabilities, which is a key aspect of probability theory, particularly in areas like gambling, investment, and finance. Let's explore the details of the problem, break down the information provided, and then calculate the probability that the price of a certain stock will go down during the next week. To achieve this, we will first define the relevant probabilities, make use of the odds provided, and ultimately calculate the probability of the stock price going down, given the constraints of the situation.

Understanding the Problem and Odds

In probability theory, odds refer to the ratio of the likelihood of an event happening to the likelihood of it not happening. These odds can be expressed as "odds in favor" or "odds against" an event, and they serve as an alternative way to represent probabilities.

The investment consultant provides two pieces of information:

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

We are tasked with determining the probability that the price of the stock will go down during the next week. To solve this, we need to consider all three possible outcomes:

1.    The stock price goes up.

2.    The stock price remains the same.

3.    The stock price goes down.

These three events are mutually exclusive and exhaustive, meaning that one of them must occur, and there is no overlap between the events. Therefore, the sum of their probabilities must equal 1.

Step 1: Converting Odds to Probabilities

Before we can calculate the probability of the stock price going down, we need to convert the given odds into probabilities. The two odds provided in the problem can be converted as follows:

Odds Against the Price Going Up: 2:1

The phrase "odds against the price going up are 2:1" means that for every 2 instances where the price does not go up, there is 1 instance where the price does go up. In terms of probabilities, this can be interpreted as follows:

• The probability that the price does go up is the ratio of favorable outcomes (1) to the total number of possible outcomes (2 + 1 = 3). P(price goes up)=13P(\text{price goes up}) = \frac{1}{3}P(price goes up)=31​
• The probability that the price does not go up (i.e., it either remains the same or goes down) is the ratio of unfavorable outcomes (2) to the total number of possible outcomes (2 + 1 = 3). P(price does not go up)=23P(\text{price does not go up}) = \frac{2}{3}P(price does not go up)=32​

Odds in Favor of the Price Remaining the Same: 1:3

The phrase "odds in favor of the price remaining the same are 1:3" means that for every 1 instance where the price remains the same, there are 3 instances where the price either goes up or goes down. In terms of probabilities, this can be interpreted as:

• The probability that the price remains the same is the ratio of favorable outcomes (1) to the total number of possible outcomes (1 + 3 = 4). P(price remains the same)=14P(\text{price remains the same}) = \frac{1}{4}P(price remains the same)=41​
• The probability that the price does not remain the same (i.e., it either goes up or goes down) is the ratio of unfavorable outcomes (3) to the total number of possible outcomes (1 + 3 = 4). P(price does not remain the same)=34P(\text{price does not remain the same}) = \frac{3}{4}P(price does not remain the same)=43​

Step 2: Defining the Events

Let us now define the events and their probabilities based on the odds provided:

• Event A: The price goes up. We have calculated: P(A)=13P(A) = \frac{1}{3}P(A)=31​
• Event B: The price remains the same. We have calculated: P(B)=14P(B) = \frac{1}{4}P(B)=41​
• Event C: The price goes down. This is the event we are trying to determine.

Step 3: Relating the Probabilities

Since the three events (price going up, price remaining the same, and price going down) are mutually exclusive and exhaustive, their probabilities must sum to 1:

P(A)+P(B)+P(C)=1P(A) + P(B) + P(C) = 1P(A)+P(B)+P(C)=1

Substituting the known probabilities for P(A)P(A)P(A) and P(B)P(B)P(B):

13+14+P(C)=1\frac{1}{3} + \frac{1}{4} + P(C) = 131​+41​+P(C)=1

To solve for P(C)P(C)P(C), the probability that the price goes down, we need to first find a common denominator between 13\frac{1}{3}31​ and 14\frac{1}{4}41​. The least common denominator of 3 and 4 is 12. So, we rewrite the fractions:

13=412,14=312\frac{1}{3} = \frac{4}{12}, \quad \frac{1}{4} = \frac{3}{12}31​=124​,41​=123​

Now, the equation becomes:

412+312+P(C)=1\frac{4}{12} + \frac{3}{12} + P(C) = 1124​+123​+P(C)=1

Simplifying the left-hand side:

712+P(C)=1\frac{7}{12} + P(C) = 1127​+P(C)=1

Now, subtract 712\frac{7}{12}127​ from both sides:

P(C)=1−712=512P(C) = 1 - \frac{7}{12} = \frac{5}{12}P(C)=1−127​=125​

Step 4: Conclusion

The probability that the price of the stock will go down during the next week is 512\frac{5}{12}125​. This is the final solution, and it represents the likelihood of the stock price experiencing a decrease, given the odds provided in the problem.

Thus, we can summarize the entire process as follows:

• We first converted the odds into probabilities.
• We then established the relationship between the probabilities of the three mutually exclusive events (price going up, remaining the same, and going down).
• Finally, we used the fact that the sum of the probabilities must equal 1 to solve for the probability of the price going down.

The final probability that the stock price will go down is 512\frac{5}{12}125​.

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