Respuesta :
Answer:
0.1733 = 17.33% probability the first stack was selected.
Step-by-step explanation:
To solve this question, it is needed to understand conditional probability, and the hypergeometric distribution.
Hypergeometric distribution:
The probability of x sucesses is given by the following formula:
[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}*C_{N-k,n-x}}{C_{N,n}}[/tex]
In which:
x is the number of sucesses.
N is the size of the population.
n is the size of the sample.
k is the total number of desired outcomes.
Combinations formula:
[tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
Conditional probability:
We use the conditional probability formula to solve this question. It is
[tex]P(B|A) = \frac{P(A \cap B)}{P(A)}[/tex]
In which
P(B|A) is the probability of event B happening, given that A happened.
[tex]P(A \cap B)[/tex] is the probability of both A and B happening.
P(A) is the probability of A happening.
Probability of all red cards for the first stack:
For this, we use the hypergeometric distribution, as the cards all chosen without replacement.
7 + 4 = 11 cards, which means that [tex]N = 11[/tex].
7 red, which means that [tex]k = 7[/tex]
3 are chosen, which means that [tex]n = 3[/tex]
We want all red, so we find P(X = 3).
[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 = 3) = h(3,11,3,7) = \frac{C_{7,3}*C_{4,0}}{C_{11,3}} = 0.2121[/tex]
Conditional probability:
Event A: All red
Event B: From the first stack.
Probability of all red cards:
0.2121 of 50%(first stack)
1 of 50%(second stack). So
[tex]P(A) = 0.2121*0.5 + 1*0.5 = 0.60605[/tex]
Probability of all red cards and from the first stack:
0.21 of 0.5. So
[tex]P(A \cap B) = 0.21*0.5 = 0.105[/tex]
What is the probability the first stack was selected?
[tex]P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.105}{0.60605} = 0.1733[/tex]
0.1733 = 17.33% probability the first stack was selected.
