La probabilidad de que un nuevo producto tenga éxito es de 0.85. Si se eligen 10 personas al azar y se les pregunta si comprarían el nuevo producto, ¿cuál es la probabilidad de que por lo menos 8 adquieran el nuevo producto?

Answer:

La probabilidad pedida es [tex]0.820196[/tex]

Explanation:

Sabemos que la probabilidad de que un nuevo producto tenga éxito es de 0.85. Sabemos también que se eligen 10 personas al azar y se les pregunta si comprarían el nuevo producto. Para responder a la pregunta, primero definiremos la siguiente variable aleatoria :

[tex]X:[/tex] '' Número de personas que adquirirán el nuevo producto de 10 personas a las que se les preguntó ''

Ahora bien, si suponemos que la probabilidad de que el nuevo producto tenga éxito se mantiene constante [tex](p=0.85)[/tex] y además suponemos que hay independencia entre cada una de las personas al azar a las que se les preguntó ⇒ Podemos modelar a [tex]X[/tex] como una variable aleatoria Binomial. Esto se escribe :

[tex]X[/tex] ~ [tex]Bi(n,p)[/tex] en donde [tex]''n''[/tex] es el número de personas entrevistadas y [tex]''p''[/tex] es la probabilidad de éxito (una persona adquiriendo el producto) en cada caso.

Utilizando los datos ⇒ [tex]X[/tex] ~ [tex]Bi(10,0.85)[/tex]

La función de probabilidad de la variable aleatoria binomial es :

[tex]p_{X}(x)=P(X=x)=\left(\begin{array}{c}n&x\end{array}\right)p^{x}(1-p)^{n-x}[/tex]    con [tex]x=0,1,2,...,n[/tex]

Si reemplazamos los datos de la pregunta en la función de probabilidad obtenemos :

[tex]P(X=x)=\left(\begin{array}{c}10&x\end{array}\right)(0.85)^{x}(0.15)^{10-x}[/tex] con [tex]x=0,1,2,...,10[/tex]

Nos piden la probabilidad de que por lo menos 8 personas adquieran el nuevo producto, esto es :

[tex]P(X\geq 8)=P(X=8)+P(X=9)+P(X=10)[/tex]

Calculando [tex]P(X=8), P(X=9)[/tex] y [tex]P(X=10)[/tex] por separado y sumando, obtenemos que [tex]P(X\geq 8)=0.820196[/tex]

Por definición de variable aleatoria binomial, la probabilidad de que por lo menos 8 adquieran el nuevo producto es 0.8202.

En primer lugar, debes saber que la Probabilidad es la mayor o menor posibilidad de que ocurra un determinado suceso.

En otras palabras, la probabilidad establece una relación entre el número de sucesos favorables y el número total de sucesos posibles.

Por otro lado, una variable aleatoria es una función que asocia un número real, perfectamente definido, a cada punto muestral.

Una variable aleatoria es discreta si los números a los que da lugar son números enteros.

Finalmente, una distribución binomial es una distribución de probabilidad discreta que describe el número de éxitos al realizar n experimentos independientes entre sí, acerca de una variable aleatoria.

La expresión  para calcular la distribución binomial es:

P(X=x)=( [tex]n\\ x[/tex] ) pˣ qⁿ⁻ˣ

Donde:

Recordar que:

( [tex]n\\ x[/tex] )= [tex]\frac{n!}{x!(n-x)!}[/tex]

donde el signo de exclamación representa el símbolo de factorial, es decir el producto de todos los números enteros positivos desde 1 hasta n.

En este caso, sabes que la probabilidad de que un nuevo producto tenga éxito es de 0.85 y se eligen 10 personas al azar. Entonces se define la siguiente variable aleatoria:

X= '' Número de personas que adquirirán el nuevo producto de 10 personas a las que se les preguntó ''

Suponiendo que hay independencia entre cada una de las personas al azar a las que se les preguntó, es posible usar una variable aleatoria Binomial. Siendo n el número de personas entrevistadas (en este caso, 10 personas) y p la probabilidad de éxito (una persona adquiriendo el producto, en este caso 0.85) en cada caso, la expresión a utilizar queda expresada como:

P(X=x)=( [tex]10\\ x[/tex] ) 0.85ˣ (1-0.85)¹⁰⁻ˣ

P(X=x)= [tex]\frac{10!}{x!(10-x)!}[/tex] 0.85ˣ (0.15)¹⁰⁻ˣ

Se desea calcular la probabilidad de que por lo menos 8 adquieran el nuevo producto. Siendo 10 las personas elegidas al azar, puede ser que sean 8,9 o 10 las personas que adquieran un nuevo producto. Entonces la probabilidad deseada es calculada como:

[tex]P(X\geq 8)= P(X=8) + P(X=9) + (P=10)[/tex]

Entonces:

[tex]P(X\geq 8)=[/tex]  [tex]\frac{10!}{8!(10-8)!}[/tex] 0.85⁸ (0.15)¹⁰⁻⁸ +  [tex]\frac{10!}{9!(10-9)!}[/tex] 0.85⁹ (0.15)¹⁰⁻⁹ +  [tex]\frac{10!}{10!(10-10)!}[/tex] 0.85¹⁰ (0.15)¹⁰⁻¹⁰

Resolviendo:

[tex]P(X\geq 8)=[/tex]  [tex]\frac{10!}{8!2!}[/tex] 0.85⁸ (0.15)² +  [tex]\frac{10!}{9!1!}[/tex] 0.85⁹ (0.15)¹ +  [tex]\frac{10!}{10!0!}[/tex] 0.85¹⁰ (0.15)⁰

[tex]P(X\geq 8)=[/tex]  45×0.85⁸×(0.15)² +  10×0.85⁹×(0.15)¹ +  1×0.85¹⁰×(0.15)⁰

[tex]P(X\geq 8)=[/tex]  0.2759 +  0.3474 + 0.1969

[tex]P(X\geq 8)=[/tex]  0.8202

Finalmente, la probabilidad de que por lo menos 8 adquieran el nuevo producto es 0.8202.

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