Respuesta :
Answer:
(1) The probability of a successful product given the product is favorable is 0.7778.
(2) The probability of a successful product given the product is unfavorable is 0.2391.
(3) The probability of a unsuccessful product given the product is favorable is 0.2222.
(4) The probability of a unsuccessful product given the product is favorable is 0.7609.
Step-by-step explanation:
Denote the events as follows:
S = a product is successful.
F = a product is favorable.
The information provided is:
[tex]P(S\cap F)=0.42\\P(S\cap F^{c})=0.11\\P(S^{c}\cap F)=0.12\\P(S^{c}\cap F^{c})=0.35\\[/tex]
The law of total probability states that:
[tex]P(A)=P(A\cap B)+P(A\cap B^{c})[/tex]
Use the law of total probability to compute the probability of a favorable product as follows:
[tex]P(F)=P(S\cap F)+P(S^{c}\cap F)\\=0.42+0.12\\=0.54[/tex]
The probability of a favorable product is 0.54.
The conditional probability of an event A given that another event B has already occurred is:
[tex]P(A|B)=\frac{P(A\cap B)}{P(B)}[/tex]
(1)
Compute the value of P (S|F) as follows:
[tex]P(S|F)=\frac{P(S\cap F)}{P(F)}=\frac{0.42}{0.54}=0.7778[/tex]
Thus, the probability of a successful product given the product is favorable is 0.7778.
(2)
Compute the value of [tex]P(S|F^{c})[/tex] as follows:
[tex]P(S|F^{c})=\frac{P(S\cap F^{c})}{P(F^{c})}=\frac{0.11}{(1-0.54)}=0.2391[/tex]
Thus, the probability of a successful product given the product is unfavorable is 0.2391.
(3)
Compute the value of [tex]P (S^{c}|F)[/tex] as follows:
[tex]P (S^{c}|F)=\frac{P(S^{c}\cap F)}{P(F)}=\frac{0.12}{0.54}=0.2222[/tex]
Thus, the probability of a unsuccessful product given the product is favorable is 0.2222.
(4)
Compute the value of [tex]P (S^{c}|F^{c})[/tex] as follows:
[tex]P (S^{c}|F)=\frac{P(S^{c}\cap F^{c})}{P(F^{c})}=\frac{0.35}{(1-0.54)}=0.7609[/tex]
Thus, the probability of a unsuccessful product given the product is favorable is 0.7609.
