Question

question 1 Consider a sample space S, and two events A and B such that P(A ∩ B) = 0.2, P(A ∪ B) = 0.6, P(B ∪ ̄A) = 0.8 (a) [0.5 points] Calculate P (A). (b) [0.5 points] Calculate P (B)

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Para calcular P(A), podemos utilizar la fórmula de inclusión-exclusión:

P(A) = P(A ∪ B) - P(B ∩ ̄A)

P(B ∩ ̄A) = P(B) - P(B ∩ A)

(a) Para calcular P(A):

P(A) = P(A ∪ B) - P(B ∩ ̄A)

P(A) = 0.6 - (P(B) - P(B ∩ A))

(b) Para calcular P(B):

P(B ∩ A) = P(A ∩ B) (por simetría)

P(B ∩ A) = 0.2

P(B ∪ ̄A) = P(B) + P( ̄A) - P(B ∩ ̄A)

0.8 = P(B) + (1 - P(A)) - 0.2

Simplificando la ecuación:

0.8 = P(B) + 1 - P(A) - 0.2

0.8 = P(B) - P(A) + 0.8

P(B) - P(A) = 0

P(B) = P(A)

Por lo tanto, P(B) = P(A).

Respondiendo a la pregunta (a):

P(A) = 0.6 - (P(B) - P(B ∩ A))

P(A) = 0.6 - (P(A) - 0.2)

Simplificando la ecuación:

P(A) = 0.4

Respondiendo a la pregunta (b):

P(B) = P(A)

P(B) = 0.4

Answer:

(a) P(A) = 0.4

(b) P(B) = 0.4

P(A) = P(A ∪ B) - P(B ∩ ̄A)

P(B ∩ ̄A) = P(B) - P(B ∩ A)

(a) Para calcular P(A):

P(A) = P(A ∪ B) - P(B ∩ ̄A)

P(A) = 0.6 - (P(B) - P(B ∩ A))

(b) Para calcular P(B):

P(B ∩ A) = P(A ∩ B) (por simetría)

P(B ∩ A) = 0.2

P(B ∪ ̄A) = P(B) + P( ̄A) - P(B ∩ ̄A)

0.8 = P(B) + (1 - P(A)) - 0.2

Simplificando la ecuación:

0.8 = P(B) + 1 - P(A) - 0.2

0.8 = P(B) - P(A) + 0.8

P(B) - P(A) = 0

P(B) = P(A)

Por lo tanto, P(B) = P(A).

Respondiendo a la pregunta (a):

P(A) = 0.6 - (P(B) - P(B ∩ A))

P(A) = 0.6 - (P(A) - 0.2)

Simplificando la ecuación:

P(A) = 0.4

Respondiendo a la pregunta (b):

P(B) = P(A)

P(B) = 0.4

Answer:

(a) P(A) = 0.4

(b) P(B) = 0.4

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