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You are dealt one card at random form a full deck and your opponent is dealt $2$ cards (Without any replacement ). If you get an Ace, he pays you $10$ dollar, if you get a King, he pays you $5$ dollar  (regardless of his cards). If you have neither an Ace nor a King, but your card is red and your opponent has no red cards, he pays you $1$ dollar. In all other cases you pay him $1$ dollar . Determine your expected earnings . Are they positive?

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The probability of getting Ace: $\frac{4}{52}$

The probability of getting King: $\frac{4}{52}$

The probability of getting Red and opponent not getting red is: $\frac{22}{52} * \frac{13*25}{25*51}$

The probability of getting Red and opponent also getting red is $\frac{22}{52} * \frac{12*25}{25*51}$+$\frac{22}{52} * \frac{25*26}{51*50}$

The probability of getting Black: $\frac{22}{52}$

 

Expectation will be: ($\frac{4}{52}*10$)+($\frac{4}{52}*5$)+($\frac{22}{52} * \frac{13*25}{25*51}*1$)+(($\frac{22}{52} * \frac{12*25}{25*51}$+$\frac{22}{52} * \frac{25*26}{51*50})*-1$)+($\frac{22}{52}*-1$)
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