There are three events A, B, and C, one of which one and only one can happen. The odds are 7 to 4 against A and 3 to 5 favour of B. Find the odds against C.

Answer

To solve the problem step by step, let's break down the information given and apply the necessary calculations.

### Step 1: Understand the Odds Against A
The odds against event A are given as 7 to 4. This means:
- For every 7 unfavorable outcomes, there are 4 favorable outcomes for A.

Using this information, we can express the probabilities:
- The total outcomes = 7 (against) + 4 (for) = 11.
- The probability of A (P(A)) can be calculated as the number of favorable outcomes divided by the total outcomes:

\[ P(A) = \frac{4}{11} \]

### Step 2: Understand the Odds in Favor of B
The odds in favor of event B are given as 3 to 5. This means:
- For every 3 favorable outcomes, there are 5 unfavorable outcomes for B.

Using this information, we can express the probabilities:
- The total outcomes = 3 (for) + 5 (against) = 8.
- The probability of B (P(B)) can be calculated as the number of favorable outcomes divided by the total outcomes:

\[ P(B) = \frac{3}{8} \]

### Step 3: Set Up the Equation for Event C
Since only one of the events A, B, or C can happen, we know that the sum of their probabilities must equal 1:

\[ P(A) + P(B) + P(C) = 1 \]

Substituting the values we found:

\[ \frac{4}{11} + \frac{3}{8} + P(C) = 1 \]

### Step 4: Find a Common Denominator
To solve for P(C), we need a common denominator for the fractions. The least common multiple of 11 and 8 is 88. We will convert both fractions:

\[ P(A) = \frac{4}{11} = \frac{4 \times 8}{11 \times 8} = \frac{32}{88} \]
\[ P(B) = \frac{3}{8} = \frac{3 \times 11}{8 \times 11} = \frac{33}{88} \]

### Step 5: Substitute and Solve for P(C)
Now substituting back into the equation:

\[ \frac{32}{88} + \frac{33}{88} + P(C) = 1 \]

Combining the fractions:

\[ \frac{32 + 33}{88} + P(C) = 1 \]
\[ \frac{65}{88} + P(C) = 1 \]

Now, isolate P(C):

\[ P(C) = 1 - \frac{65}{88} = \frac{88 - 65}{88} = \frac{23}{88} \]

### Step 6: Find the Odds Against C
The odds against event C can be calculated using the formula:

\[ \text{Odds against C} = \frac{P(C')}{P(C)} \]

Where P(C') is the probability of not C:

\[ P(C') = 1 - P(C) = 1 - \frac{23}{88} = \frac{65}{88} \]

Now substituting into the odds formula:

\[ \text{Odds against C} = \frac{P(C')}{P(C)} = \frac{65/88}{23/88} = \frac{65}{23} \]

### Final Answer
Thus, the odds against event C are:

\[ \text{Odds against C} = 65 : 23 \]