Given: \[\begin{align*} P(A) &= \frac{5}{8}, \\ P(F) &= \frac{5}{8}, \\ P(A \cap F) &= \frac{1}{4}. \end{align*}\]
To find \(P(M)\): \[\begin{align*} P(M) &= 1 - (P(A) + P(F) - P(A \cap F)) \\ &= 1 - \left(\frac{5}{8} + \frac{5}{8} - \frac{1}{4}\right) \\ &= 0. \end{align*}\]
Probability of choosing Mathematics (\(P(M)\)): - The probability of choosing Mathematics is \(0\). This means that it’s impossible for the student to choose Mathematics as one of their electives given the provided probabilities of choosing Art (\(P(A)\)), French (\(P(F)\)), and both Art and French together (\(P(A \cap F)\)). 2.
To find \(P(A \cup F)\): \[\begin{align*} P(A \cup F) &= P(A) + P(F) - P(A \cap F) \\ &= \frac{5}{8} + \frac{5}{8} - \frac{1}{4} \\ &= 1. \end{align*}\]