Right Strategy for a Weaker Player

The strategy is to play a daring game when tied or losing and a conservative game when ahead in the match.

After two games their are four possible outcomes (from A's perspective):

$\begin{array}{c|c|c} \text{ result }&\text{ conclusion }&\text{ probability }\\ \hline LL&\text{ match lost }&0.55\cdot 0.55=0.3025\\ LW&\text{ tie-breaker required }& 0.55\cdot 0.45=0.2475\\ WL&\text{ tie-breaker required }& 0.45\cdot 0.10=0.0450\\ WD&\text{ match won }&0.45\cdot 0.90=0.4050 \end{array}$

The tie-breaker happens with the probability of $0.2925$ so A is going to play a daring game, winning with the probability of $0.45,$ making the total probability of his winning the match equal to

$0.4050+0.45\cdot 0.2925=0.536625\approx 54\%.$

The problem and the solution are from J. G. McLoughlin et all, Jim Totten's Problems of the Week, World Scientific, 2013, problem #316.

Jakub Šimek made this remark: One should not underestimate the ability to draw under all circumstances. Also, the oponent style makes the strategy possible, she cannot play to draw or something... Really interesting...