Answer
-11.76 dollars
Step-by-step explanation
First, we need to calculate the probability of winning, that is, the probability of drawing three black cards in succession without replacement.
In the beginning, there are a total of 52 cards, and 26 of them are black, then the probability that the first card drawn is black is:
[tex]P(first\text{ card is black})=\frac{26}{52}=\frac{1}{2}[/tex]After drawing 1 black card, there are a total of 51 cards, and 25 of them are black, then the probability that the second card drawn is black is:
[tex]P(\text{ second card is black})=\frac{25}{51}[/tex]After drawing 2 black cards, there are a total of 50 cards, and 24 of them are black, then the probability that the third card drawn is black is:
[tex]P(\text{ third card is black})=\frac{24}{50}=\frac{12}{25}[/tex]Therefore, the probability of winning, that is, the three cards are black is calculated as follows:
[tex]\begin{gathered} P(\text{ first card is black AND second card is black AND third card is black})=P(\text{ first card is black})\cdot P(\text{ second card is black })\cdot P(\text{ third card is black }) \\ P(winning)=\frac{1}{2}\cdot\frac{25}{51}\cdot\frac{12}{25} \\ P(w\imaginaryI nn\imaginaryI ng)=\frac{6}{51} \end{gathered}[/tex]The probability of losing is the complement of the probability of winning, that is,
[tex]\begin{gathered} P(\text{ losing})=1-P(winning) \\ P(\text{los}\imaginaryI\text{ng})=1-\frac{6}{51} \\ P(\text{los}\mathrm{i}\text{ng})=\frac{45}{51} \end{gathered}[/tex]Finally, the expected value is calculated as follows:
[tex]\begin{gathered} EV=P(winning)\cdot\text{ Amount won per bet}-P(losing)\cdot\text{ Amount lost per bet} \\ EV=\frac{6}{51}{}\cdot\text{ \$}50\text{ - }\frac{45}{51}\cdot\text{ \$}20 \\ EV=-\text{ \$}11.76 \end{gathered}[/tex]