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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: ($\frac{4}{52}*10$)+($\frac{4}{52}*5$)+($\frac{22}{52} * \frac{13*25}{25*51}*1$)+(($\frac{22}{52} * \frac{12*25}{25*51}$+$\frac{22}{52} * \frac{25*26}{51*50})*-1$)+($\frac{22}{52}*-1$)
by Boss (18.5k points)
edited by
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The probability of getting Red and opponent also getting red is

why? plz explain?

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i am getting RED ( but not King or ACE ) ====>  Probability = $\frac{(26-4)}{(52)}$ , opposite person should not have red cards = Probability = $\frac{\binom{26}{2}}{\binom{51}{2}}$

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I have taken two cases in case of opponent also getting red

1) is both the cards are red

2) only one is red
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Do tell me if I missed something or if I am wrong
+1
what i want to convey is, for opponent you have 51 cards but not 52.