Probability of selecting a red number given that the circle is yellow?

If "circle is yellow," there is no probability of getting a red number.

To calculate the probability of selecting a red number given that the circle is yellow, we need to have some additional information about the problem. Specifically, we need to know how many numbers are in the circle and how many of them are red.

Once we have this information, we can use the formula for conditional probability:

\[ P(\text{red}|\text{yellow}) = \frac{P(\text{red and yellow})}{P(\text{yellow})} \]

where \( P(\text{red}|\text{yellow}) \) represents the probability of selecting a red number given that the circle is yellow.

To find the probability of red and yellow together, we need to know the number of red circles. Let's call that \( n_{\text{red}} \), and the number of yellow circles \( n_{\text{yellow}} \). Then, the probability of red and yellow together is given by:

\[ P(\text{red and yellow}) = \frac{n_{\text{red}}}{n_{\text{total}}} \times \frac{n_{\text{yellow}}}{n_{\text{total}}} \]

Next, we need to determine the probability of the circle being yellow, denoted as \( P(\text{yellow}) \). If we assume that each circle has an equal chance of being selected, then the probability of selecting a yellow circle would be:

\[ P(\text{yellow}) = \frac{n_{\text{yellow}}}{n_{\text{total}}} \]

Finally, we substitute these values into the formula for conditional probability:

\[ P(\text{red}|\text{yellow}) = \frac{\frac{n_{\text{red}}}{n_{\text{total}}} \times \frac{n_{\text{yellow}}}{n_{\text{total}}}}{\frac{n_{\text{yellow}}}{n_{\text{total}}}} \]

Simplifying this expression gives us the probability of selecting a red number given that the circle is yellow.