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# 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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## Answer to a math 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 $B$

Murray
4.5
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

$a$ P$A$ = 0.4
$b$ P$B$ = 0.4
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