Let's solve this problem step-by-step:

Given:

• The probability of winning the game is $\frac{6}{10}$.
• The question asks for the odds of losing a game.

Step 1: Calculate the probability of losing the game.

The probability of losing the game is the complement of the probability of winning, which means: $P(\text{losing}) = 1 - P(\text{winning}) = 1 - \frac{6}{10} = \frac{4}{10}$

Step 2: Determine the odds of losing the game.

The odds are given as the ratio of losses to wins. In this case:

• The probability of losing is $\frac{4}{10}$.
• The probability of winning is $\frac{6}{10}$.

The odds of losing is the ratio of losses to wins, which is: $\text{Odds of losing} = \frac{\text{losses}}{\text{wins}} = \frac{4}{6}$

Simplifying the fraction $\frac{4}{6}$ gives $\frac{2}{3}$.

Step 3: Match this with the provided options.

The ratio $4:6$ is one of the available choices in the image. Therefore, the correct answer is 4:6.

Would you like more details on the solution, or do you have any further questions?

Here are 5 related questions to expand on this:

1. What are the odds of winning the game based on the same probabilities?
2. If the probability of winning changes to $\frac{7}{10}$, what are the new odds of losing?
3. How do you convert probabilities into odds for general cases?
4. What is the difference between probability and odds?
5. How would the odds change if the game were played multiple times with the same probabilities?

Tip: Always remember that odds and probabilities, though related, are not the same. Probabilities focus on outcomes relative to the total, while odds compare favorable and unfavorable outcomes directly.