Let's solve the problem step by step.
1. Total Number of Candidates
There are 4 females and 3 males, summing up to:
[tex]\[
\text{Total candidates} = 4 + 3 = 7
\][/tex]
2. Probability of Electing a Female President
The probability of electing a female president is given by the number of females divided by the total number of candidates:
[tex]\[
P(\text{Female President}) = \frac{\text{Number of females}}{\text{Total candidates}} = \frac{4}{7} \approx 0.5714
\][/tex]
3. Remaining Candidates After Electing Female President
After a female is elected president, there are 6 candidates left (4 + 3 - 1):
[tex]\[
\text{Remaining candidates} = 7 - 1 = 6
\][/tex]
4. Probability of Electing a Male Vice President Given a Female President
There are still 3 males left out of the remaining 6 candidates. Thus, the probability of electing a male vice president given that a female president has been elected is:
[tex]\[
P(\text{Male Vice President} | \text{Female President}) = \frac{\text{Number of males}}{\text{Remaining candidates}} = \frac{3}{6} = 0.5
\][/tex]
5. Combined Probability of Electing a Female President and a Male Vice President
To find the combined probability of both events happening, we multiply the probabilities of the two independent events:
[tex]\[
P(\text{Female President and Male Vice President}) = P(\text{Female President}) \times P(\text{Male Vice President} | \text{Female President}) = \frac{4}{7} \times \frac{3}{6} = \frac{4}{7} \times 0.5 = \frac{2}{7} \approx 0.2857
\][/tex]
Hence, the correct answer is:
[tex]\[
\boxed{\frac{2}{7}}
\][/tex]
