2025 Fermat Problem 17

A tennis tournament starts with 8 players. Francesca is equally likely to play against any of the other 7 players in her first match. If Francesca plays against Dominique or Estella, the probability that Francesca wins is \frac{2}{5}. If Francesca plays against any of the other 5 players, the probability that she wins is \frac{3}{4}. What is the probability that Francesca wins her first match?

\textbf{(A)}\ \frac{23}{50}\quad \textbf{(B)}\ \frac{29}{50}\quad \textbf{(C)}\ \frac{3}{5}\quad \textbf{(D)}\ \frac{13}{20}\quad \textbf{(E)}\ \frac{2}{3}

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Full credit to this problem is given to the CEMC, you may view all Fermat contests here.