Consider a box that contains 10 red balls and 8 blue balls. Alex draws a ball fr

Question

Consider a box that contains 10 red balls and 8 blue balls. Alex draws a ball from the box and shows it to his friend Joe at some distance from the site. Unfortunately, Joe is viewing the balls over a fuzzy environment with the result that if the ball drawn is a red ball, he will say that it is a red ball with probability 0.8 and that it is a blue ball with probability 0.2. Similarly, if the ball drawn is a blue ball he will say that it is a red ball with probability 0.3 and that it is blue ball with probability 0.7. The balls are drawn with replacement. Given that Joe says that the ball drawn is a red ball, what is the probability that it is actually a red ball? Given that Joe says that the ball drawn is a blue ball, what is the probability that it is actually a red ball? What is the probability that Joe makes a mistake in identifying an arbitrary ball? Angela is taking a multiple-choice exam in which each question has 4 possible answers. She knows the correct answers to 40% of the questions. In 40% of the questions she can narrow the choices down to 2 answers one of which is the correct answer, which means that in this case she will be correct 50% of the time. She knows nothing about the remaining 20% of the questions and will equally choose one of them. What is the probability that Angela will correctly answer a question chosen at random from the test? Given that Angela answered a question correctly, what is the probability that it was a question she absolutely knows nothing about? (Note that we asking for the probability that the question is one of those that she randomly chose one of the 4 answers, given that he is correct.)

Explanation / Answer

7)here let probabilty of blue ball =P(B)=8/18

and probability of red ball=P(R) =10/18

also probabilty that Joe see red Ball=P(r)

and he sees blue ball =P(b)

a) probabilty that ball seen is red =P(r)=P(R)*P(r|R)+P(B)*P(r|B) =(10/18)*0.8+(8/18)*0.3 =0.5778

probability that it is red ball, given ball seen is red =P(R|r)=P(R)*P(r|R)/P(r)=(10/18)*0.8/0.5778=0.7692

b)probabilty of blue ball P(b)=P(R)*P(b|R)+P(B)*P(b|B)=(10/18)*0.2+(8/18)*0.7=0.4222

hence probabilty of red ball, given seen blue ball=P(R|b) =P(R)*P(b|R)/P(b) =(10/18)*0.2/0.42222=0.2632

c)probabilty that Joe makes a mistake =P(r|B)+P(b|R) =(10/18)*0.2+(8/18)*0.3=0.2444

8)let probabilty of answering correct =P(C)

and probabilty it is from 1st type =P(1); from second=P(2);from 3rd=P(3)

hence a) answering a question correct =P(C)=P(1)*P(C|1)+P(2)*P(C|2)+P(3)*P(C|3)=0.4*1+0.4*0.5+0.2*0.25

=0.65

b)probabilty that it is from category 3 ; given she correctly answers it =P(3|C)=P(3)*P(C|3)/P(C)=0.2*0.25/0.65

=0.0769

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