The odds in favour of A winning a game of chess against B are 5:2. if three games are to be played, then the odds in favour of A's winning at least one game are 355:8.

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the odds in favor of A winning
The odds in favor of A winning against B are given as 5:2. This means that for every 5 times A wins, B wins 2 times.

**Hint:** Convert the odds into probabilities. The probability of A winning (P(A)) can be calculated as:
\[ P(A) = \frac{\text{Odds in favor of A}}{\text{Total Odds}} = \frac{5}{5+2} = \frac{5}{7} \]

### Step 2: Calculate the probability of A losing
If A has a probability of winning, then the probability of A losing (P(B)) can be calculated as:
\[ P(B) = 1 - P(A) = 1 - \frac{5}{7} = \frac{2}{7} \]

**Hint:** Remember that the total probability must sum to 1.

### Step 3: Calculate the probability of A losing all three games
If three games are played, the probability of A losing all three games is:
\[ P(\text{A loses all 3}) = P(B)^3 = \left(\frac{2}{7}\right)^3 = \frac{8}{343} \]

**Hint:** Use the multiplication rule for independent events.

### Step 4: Calculate the probability of A winning at least one game
The probability of A winning at least one game is the complement of A losing all three games:
\[ P(\text{A wins at least 1}) = 1 - P(\text{A loses all 3}) = 1 - \frac{8}{343} = \frac{343 - 8}{343} = \frac{335}{343} \]

**Hint:** Use the complement rule to find the probability of at least one success.

### Step 5: Convert the probability to odds
Now, we need to convert the probability of A winning at least one game into odds. The odds in favor of A winning at least one game can be calculated as:
\[ \text{Odds in favor of A winning at least one} = \frac{P(A \text{ wins at least 1})}{P(A \text{ loses at least 1})} \]
Where \( P(A \text{ loses at least 1}) = 1 - P(A \text{ wins at least 1}) = \frac{8}{343} \).

Calculating the odds:
\[ \text{Odds} = \frac{\frac{335}{343}}{\frac{8}{343}} = \frac{335}{8} \]

**Hint:** Remember that odds are expressed as a ratio of two probabilities.

### Final Result
Thus, the odds in favor of A winning at least one game out of three is \( 335:8 \).