<ul><li>Games extending to mixed strategies allow players to choose actions with given probabilities.</li><li>Probability in games enables multiple plays and consideration of players' action likelihoods.</li><li>Nash equilibrium determines optimal strategies by considering expected rewards.</li><li>In the soccer game example, finding the Nash equilibrium resulted in optimal corner selection probabilities.</li><li>Players' assumptions about opponents' actions influence their own strategies in game theory.</li><li>Real-world scenarios like crime reporting can be analyzed using game theory concepts.</li><li>Calculating expected rewards helps determine equilibrium in complex situations with multiple players.</li><li>Rational behavior involves maximizing expected rewards based on others' mixed action strategies.</li><li>As the number of witnesses to a crime increases, the likelihood of individual reporting decreases.</li><li>Effortless and low-threshold reporting mechanisms increase the probability of crime reporting.</li></ul>

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