Question

You are dealt one card at random from a full deck and your opponent is dealt 2 cards (without any replacement). If you get an Ace, he pays you $10, if you get a King, he pays you $5 (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. In all other cases you pay him $1. Determine your expected earnings. Are they positive?

Answer 1

Suppose that a card deck contains $52$ cards. The probability of getting an Ace is: $p_1=\frac{4}{52}=\frac{1}{13}$. The probability of getting a King is: $p_2=\frac{4}{52}=\frac{1}{13}$. The probability of getting a red card, which is neither an Ace nor a King is: $p_3=\frac{11}{52}$. The probability that the opponent has no red cards is: $p_4=\frac{39}{52}$. The expected earning is: $E[X]=10p_1+5p_2+p_3p_4-(1-p_1-p_2-p_3p_4)=\frac{10}{13}+\frac{5}{13}+\frac{11}{52}\cdot\frac{39}{52}-(1-\frac{2}{13}-\frac{11}{52}\cdot\frac{39}{52})=\frac{5}{8}>0$

$X$ denotes a random variable that corresponds to earnings. The expected earning is positive. The expected earning is equal to $\frac{5}{8}.$