Problem 562

The probability that Francesca wins her first game is equal to
the probability that she plays either Dominique or Estella and wins,
added to the probability that she plays any of the other $5$ players and wins.

It is equally likely that Francesca plays her first game against any of
the other $7$ players, and so the
probability that she plays against either Dominique or Estella is $\dfrac27$.

The probability that she wins her first game when playing either of
these two players is $\dfrac25$, and
so the probability that she plays her first match against Dominique or
Estella and wins is $\dfrac27\times\dfrac25=\dfrac{4}{35}$.

The probability that Francesca plays her first game against any of the
other $5$ players is $\dfrac57$.

The probability that she wins her first game when playing any of the
other $5$ players is $\dfrac34$, and so the probability that she
plays her first match against any of the other $5$ players and wins is $\dfrac57\times\dfrac34=\dfrac{15}{28}$.

Thus, the probability that Francesca wins her first match is $\dfrac{4}{35}+\dfrac{15}{28}=\dfrac{16}{140}+\dfrac{75}{140}=\dfrac{91}{140}=\dfrac{13}{20}$.