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 gives us the following odds:

1.     Odds against the stock price going up: 2:1

o    This means the probability that the stock will go up is 13\frac{1}{3}31​ and the probability that it will not go up is 23\frac{2}{3}32​.

2.     Odds in favor of the stock price remaining the same: 1:3

o    This means the probability that the stock will remain the same is 14\frac{1}{4}41​ and the probability that it will not remain the same is 34\frac{3}{4}43​.

To find the probability that the stock price will go down, we can use the fact that the three possible outcomes (price going up, staying the same, or going down) are mutually exclusive and collectively exhaustive events, meaning their probabilities add up to 1.

Let:

• P(up)P(\text{up})P(up) be the probability that the price goes up.
• P(same)P(\text{same})P(same) be the probability that the price stays the same.
• P(down)P(\text{down})P(down) be the probability that the price goes down.

Given:

• P(up)=13P(\text{up}) = \frac{1}{3}P(up)=31​
• P(same)=14P(\text{same}) = \frac{1}{4}P(same)=41​

We can calculate P(down)P(\text{down})P(down) as follows:

P(down)=1−P(up)−P(same)P(\text{down}) = 1 - P(\text{up}) - P(\text{same})P(down)=1−P(up)−P(same)

Substituting the values:

P(down)=1−13−14P(\text{down}) = 1 - \frac{1}{3} - \frac{1}{4}P(down)=1−31​−41​

To perform this calculation, we first find a common denominator:

P(down)=1−412−312=1−712=512P(\text{down}) = 1 - \frac{4}{12} - \frac{3}{12} = 1 - \frac{7}{12} = \frac{5}{12}P(down)=1−124​−123​=1−127​=125​

Answer:

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

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