The chance of an event happening is the square of the chance of a second event but the odds against the first are the cube of the odds against the second.

Options

A. `P_(1) = 1//9`
B. `P_(1) = 1//16`
C. `P_(2) = 1//3`
D. `P_(2) = 1//4`

Explanation

To solve the problem step by step, we will define the probabilities and odds as given in the question and then manipulate the equations accordingly.

### Step-by-Step Solution:

1. **Define the Probabilities:**
Let the probability of the second event happening be \( P \). According to the problem, the probability of the first event happening is the square of the probability of the second event:
\[ P(A) = P^2 \]

2. **Define the Odds:**
The odds against the second event happening can be defined as:
\[ \text{Odds against second event} = \frac{1 - P}{P} \]
The odds against the first event happening is given to be the cube of the odds against the second event:
\[ \text{Odds against first event} = \left(\frac{1 - P}{P}\right)^3 \]

3. **Set Up the Equation:**
The odds against the first event can also be expressed in terms of its probability:
\[ \text{Odds against first event} = \frac{1 - P^2}{P^2} \]
Therefore, we can set up the following equation:
\[ \frac{1 - P^2}{P^2} = \left(\frac{1 - P}{P}\right)^3 \]

4. **Cross-Multiply:**
Cross-multiplying gives:
\[ (1 - P^2) \cdot P^3 = (1 - P)^3 \]

5. **Expand Both Sides:**
Expanding both sides:
\[ P^3 - P^5 = 1 - 3P + 3P^2 - P^3 \]

6. **Rearranging the Equation:**
Rearranging the equation gives:
\[ P^5 - 2P^3 + 3P - 1 = 0 \]

7. **Finding Roots:**
To find the roots of this polynomial, we can try substituting \( P = \frac{1}{3} \):
\[ \left(\frac{1}{3}\right)^5 - 2\left(\frac{1}{3}\right)^3 + 3\left(\frac{1}{3}\right) - 1 = 0 \]
Simplifying:
\[ \frac{1}{243} - \frac{2}{27} + 1 - 1 = 0 \]
This confirms that \( P = \frac{1}{3} \) is indeed a root.

8. **Finding \( P^2 \):**
Now, we can find \( P^2 \):
\[ P^2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \]

### Final Answer:
The value of \( P \) is \( \frac{1}{3} \) and the value of \( P^2 \) is \( \frac{1}{9} \).