Question

The concert pianist “Donna Prima” is very upset by the number of sneezes that occur in the audience just before he begins to play. During her last tour, Donna estimated an average of 10 sneezes just before to start their concerts. Mrs. Donna Prima has warned her director that, If you hear more than 15 sneezes before starting that night's concert, you will be will refuse to touch. What is the probability that the artist plays the piano in the concert that night?

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Answer to a math question The concert pianist “Donna Prima” is very upset by the number of sneezes that occur in the audience just before he begins to play. During her last tour, Donna estimated an average of 10 sneezes just before to start their concerts. Mrs. Donna Prima has warned her director that, If you hear more than 15 sneezes before starting that night's concert, you will be will refuse to touch. What is the probability that the artist plays the piano in the concert that night?

Expert avatar
Murray
4.5
89 Answers
Para resolver este problema, necesitamos utilizar la distribución de Poisson, ya que estamos hablando de eventos discretos (número de estornudos) en un intervalo continuo (antes de iniciar el concierto).

La fórmula de la distribución de Poisson es:

P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}

Donde:
- P(X = k) es la probabilidad de que ocurran k eventos.
- \lambda es el valor medio de eventos que se espera que ocurran en un intervalo dado.
- k es el número de eventos que nos interesa.

En este caso, el valor medio de estornudos antes de iniciar el concierto es 10, por lo que \lambda = 10 . La artista se rehusará a tocar si escucha más de 15 estornudos, por lo que necesitamos encontrar la probabilidad de que haya 15 o menos estornudos:

P(X \leq 15) = P(X=0) + P(X=1) + \dots + P(X=15)

Calculamos la probabilidad para cada k y sumamos:

P(X=0) = \frac{e^{-10} \cdot 10^0}{0!}
P(X=1) = \frac{e^{-10} \cdot 10^1}{1!}
\vdots
P(X=15) = \frac{e^{-10} \cdot 10^{15}}{15!}

Finalmente sumamos todos los valores obtenidos para encontrar la probabilidad de que la artista toque el piano en el concierto de esa noche.

P(X \leq 15) = \sum_{k=0}^{15} \frac{e^{-10} \cdot 10^k}{k!}

P(X \leq 15) \approx 0.9973

Entonces, la probabilidad de que la artista toque el piano en el concierto de esa noche es aproximadamente 0.9973.

\boxed{P(X \leq 15) \approx 0.9973}

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