• The expected value is the sum of each outcome multiplied by it's probability.
• The cards are drawn without replacement, thus, the ,hypergeometric distribution, is used to solve this question.
Hypergeometric distribution:
[tex]C_{n,x}=\frac{n!}{x!(n-x)!}[/tex]The parameters are:
• x is the number of successes.
,• N is the size of the population.
,• n is the size of the sample.
,• k is the total number of desired outcomes.
In this problem:
• A deck has 52 cards, thus N = 52
,• 13 of those are spades, thus k = 13
,• 2 cards are going to be drawn, thus n = 2
,• We want 2 spades, thus, P(X = 2).
The probabilities of each outcome for your bet are:
• 0.0588 probability of earning $23
,• 0.9412 probability of losing $4
Thus, the expected value is:
(0.0588)(23) - (0.9412)(4) = - $2.41
If this same bet is made 843 times:
- $2.41 x 843 = -$2033.65
