Respuesta :
Answer:
The expected value is $-2.41.
Step-by-step explanation:
Consider the provided information.
There are 52 cards in a deck out of which 13 are spades.
The probability that both cards are spade is
[tex]\frac{13}{52} \times \frac{12}{51}=0.25 \times 0.235 = 0.0588[/tex]
If both cards are spades, your friend will pay you $39. Therefore,
[tex]0.0588(39)=2.2932[/tex]
The probability that both cards are not spade is:
[tex]1-0.0588=0.9412[/tex]
If both cards are not spades then you have to pay your friend $5. Therefore,
[tex]0.9412(-5)=-4.706[/tex]
Here, negative sign represents the losses.
The expected value of your bet is:
2.2932-4.706 = -2.4128
Hence, the expected value is $-2.41.
Using the hypergeometric distribution, it is found that the expected value of your bet is of -$2.41.
- 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]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}*C_{N-k,n-x}}{C_{N,n}}[/tex]
[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 [tex]N = 52[/tex].
- 13 of those are spades, thus [tex]k = 13[/tex].
- 2 cards are going to be drawn, thus [tex]n = 2[/tex].
- We want 2 spades, thus, P(X = 2).
[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}*C_{N-k,n-x}}{C_{N,n}}[/tex]
[tex]P(X = 2) = h(2,52,2,13) = \frac{C_{13,2}*C_{39,0}}{C_{52,2}} = 0.0588[/tex]
The probabilities of each outcome for your bet are:
- 0.0588 probability of earning $39.
- 0.9412 probability of losing $5.
Thus, the expected value is:
[tex]E(X) = 0.0588(39) - 0.9412(5) = -2.41[/tex]
The expected value of your bet is of -$2.41.
A similar problem is given at https://brainly.com/question/24855677
