Respuesta :
Answer:
0.0005 = 0.05% probability that all the cards are spades.
Step-by-step explanation:
The cards are chosen without replacement, which means that the hypergeometric distribution is used to solve this question.
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]
In this question:
52 cards means that N = 52.
13 of the cards are spades, which means that k = 13.
5 card hand means that n = 5.
What is the probability that all the cards are spades?
This is P(X = 5). So
[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 = 5) = h(5,52,5,13) = \frac{C_{13,5}*C_{39,0}}{C_{52,5}} = 0.0005[/tex]
0.0005 = 0.05% probability that all the cards are spades.
