Respuesta :
Answer:
(A) The correct option is (A).
(B) The correct option is (E).
Step-by-step explanation:
The events can be defined as follows:
X = students felt they learned better at home
Y = students plan on taking an online course in college
The information provided is:
P (X) = 0.24
P (Y|X) = 0.80
P (Y|X') = 0.40
[tex]P(Y'|X)=1-P(Y|X)\\=1-0.80\\=0.20[/tex]
[tex]P(Y'|X')=1-P(Y|X')\\=1-0.40\\=0.60[/tex]
The Bayes' theorem states that the conditional probability of an event E[tex]_{i}[/tex] given that another event A has already occurred is:
[tex]P(E_{i}|A)=\frac{P(A|E_{i})P(E_{i})}{\sum {P(A|E_{i})P(E_{i})}}[/tex]
(A)
Compute the probability a person who does not plan on taking an online course felt they learned better at home as follows:
Use the Bayes' theorem.
[tex]P(X|Y')=\frac{P(Y'|X)P(X)}{P(Y'|X)P(X)+P(Y'|X')P(X')}[/tex]
[tex]=\frac{0.20\times 0.24}{(0.20\times 0.24)+(0.60\times 0.76)}\\\\=0.09524\\\\\approx 0.095[/tex]
Thus, the probability a person who does not plan on taking an online course felt they learned better at home is 0.095 or 2/21.
(B)
Compute the probability a person who does plan on taking an online course felt they did not learn better at home as follows:
[tex]P(X'|Y')=1-P(X|Y')\\=1-0.095\\=0.905[/tex]
Thus, the probability a person who does plan on taking an online course felt they did not learn better at home is 0.905.
