Answer:
[tex]Amount = -\$1461.9[/tex]
Step-by-step explanation:
We have the following:
[tex]Total = 52[/tex] --- cards in a standard deck
Represent cards with a value of 5 or less with x
So: [tex]x = \{2, 3, 4, 5\}[/tex] i.e 4 cards
However, each of these has a frequency of 4.
So: [tex]n(x) = 4\ cards* 4[/tex]
[tex]n(x) = 16[/tex]
The probability is:
[tex]p(x) = \frac{16}{52}[/tex]
The cost of this is:
[tex]C(x) = +\$7[/tex]
Represent cards with a value of above 5 with y
[tex]n(y) = 52 - 16[/tex]
[tex]n(y) = 36[/tex]
The probability is:
[tex]p(y) = \frac{36}{52}[/tex]
The cost of this is:
[tex]C(y) = -\$7[/tex] --- It is negative because you lost
In a game, your expected amount is:
[tex]E = n(x) * C(x) + n(y) * C(y)[/tex]
[tex]E = \frac{16}{52} * (+\$7) + \frac{36}{52} * (-\$7)[/tex]
[tex]E = \frac{16* \$7}{52} - \frac{36* \$7}{52}[/tex]
[tex]E = \frac{\$112}{52} - \frac{\$252}{52}[/tex]
[tex]E = \frac{\$112 - \$252}{52}[/tex]
[tex]E = -\frac{\$140}{52}[/tex]
When you play 543 times, the expected amount is:
[tex]Amount = E * 543[/tex]
[tex]Amount = -\frac{\$140}{52} *543[/tex]
[tex]Amount = -\frac{\$140 *543}{52}[/tex]
[tex]Amount = -\frac{\$76020}{52}[/tex]
[tex]Amount = -\$1461.9[/tex]
