HolaMarianis9182!
Analizando y Resolviendo el problema :
Las probabilidades previas de los eventos A1, A2, A3 son :
P(A1) = 0.
20
P(A2) = 0.
50
P(A3) = 0.
30
Las probabilidades condicionales del evento B dados los eventos A1, A2 y A3 son :
P(B|A1) = 0.
50
P(B|A2) = 0.
40
P(B|A3) = 0.
30
a.
Calcule P(B∩A1), P(B∩A2) y P(B∩A3).
- P(B∩A1) = P(B|A1)xP(A1) = 0.
50x0.
20 = 0.
1 - P(B∩A2) = P(B|A2)xP(A2) = 0.
40x0.
50 = 0.
2 - P(B∩A3) = P(B|A3)xP(A3) = 0.
30x0.
30 = 0.
09
b.
Emplee la teorema de Bayes, para calcular la probabilidad posterior P(A2|B).
- <img src="https://tex.z-dn.net/?f=P%28A2%7CB%29%3D%20%20%5Cfrac%7BP%28B%7CA2%29%2AP%28A2%29%7D%7BP%28B%7CA1%29%2AP%28A1%29%2BP%28B%7CA2%29%2AP%28A2%29%2BP%28B%7CA3%29%2AP%28A3%29%7D%20" /> - <img src="https://tex.z-dn.net/?f=P%28A2%7CB%29%3D%20%20%5Cfrac%7B0.4%2A0.5%7D%7B%280.5%2A0.2%29%2B%280.4%2A0.5%29%2B%280.3%2A0.3%29%7D%3D%20%20%5Cfrac%7B0.2%7D%7B0.1%2B0.2%2B0.09%7D%3D%20%20%5Cfrac%7B0.2%7D%7B0.39%7D%3D%20%5Cfrac%7B20%7D%7B39%7D" /> - <img src="https://tex.z-dn.net/?f=P%28A2%7CB%29%3D%5Cfrac%7B20%7D%7B39%7D" />
c.
Use el metodo tabular para emplear el teorema de bayes en el calculo de P(A1|B), P(A2|B) y P(A3|B).
- <img src="https://tex.z-dn.net/?f=P%28A1%7CB%29%3D%20%5Cfrac%7B0.2%2A0.5%7D%7B%280.5%2A0.2%29%2B%280.4%2A0.5%29%2B%280.3%2A0.3%29%7D%3D%20%5Cfrac%7B0.1%7D%7B0.1%2B0.2%2B0.09%7D%3D%20%5Cfrac%7B0.1%7D%7B0.39%7D%3D%20%5Cfrac%7B10%7D%7B39%7D" /> - <img src="https://tex.z-dn.net/?f=P%28A3%7CB%29%3D%20%5Cfrac%7B0.3%2A0.3%7D%7B%280.5%2A0.2%29%2B%280.4%2A0.5%29%2B%280.3%2A0.3%29%7D%3D%20%5Cfrac%7B0.09%7D%7B0.1%2B0.2%2B0.09%7D%3D%20%5Cfrac%7B0.09%7D%7B0.39%7D%3D%20%5Cfrac%7B3%7D%7B13%7D" />
Espero haberte ayudado!