As part of a street performance, Manuel guesses 5 audience members’ favorite color. Each audience member may select from 4 possible colors, and must select only one. If Manuel independently and randomly guesses each person’s favorite color, what is the probability that he will guess all 5 favorite colors correctly?
Question
As part of a street performance, Manuel guesses 5 audience members’ favorite color. Each audience member may select from 4 possible colors, and must select only one. If Manuel independently and randomly guesses each person’s favorite color, what is the probability that he will guess all 5 favorite colors correctly?
Solution
Para calcular la probabilidad de que Manuel adivine correctamente el color favorito de las 5 personas, primero debemos entender que cada intento de adivinar es un evento independiente. Esto significa que la probabilidad de adivinar correctamente el color favorito de una persona no afecta la probabilidad de adivinar correctamente el color favorito de otra persona.
Cada persona tiene 4 colores posibles para elegir, por lo que la probabilidad de que Manuel adivine correctamente el color favorito de una persona es de 1 en 4, o 0.25.
Dado que estos son eventos independientes, podemos multiplicar las probabilidades individuales para obtener la probabilidad total. Entonces, la probabilidad de que Manuel adivine correctamente el color favorito de las 5 personas es de (0.25) ^ 5 = 0.0009765625.
Por lo tanto, la probabilidad de que Manuel adivine correctamente el color favorito de las 5 personas es muy baja, aproximadamente 0.0009765625 o 0.09765625%.
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