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Probability

Long Answer Type

901. There are two bags I and II. Bag I contains 3 white and 2 red balls and Bag II contains 5 white and 4 red balls. One ball is drawn at random from one of the bags and is found to be red. Find the probability that it was drawn from bag II.

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902. There are two bags I and II. Bag I contains 4 white and 3 red balls and Bag II contains 6 white and 5 red balls. One ball is drawn at random from one of the bags and is found to be red. Find the probability that it was drawn from bag II.

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903. There are two bags I and II. Bag I contains 2 white and 4 red balls and Bag II contains 5 white and 3 red balls. One ball is drawn at random from one of the bags and is found to be red. Find the probability that it was drawn from bag II.

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904. Bag I contains 3 red and 4 black balls and bag II contains 4 red and 5 black balls. One ball is transferred from bag I to bag II, then a ball is drawn from bag II, the ball so drawn is found to be red in colour. Find the probability that the transferred ball is black.

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Short Answer Type

905. A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls. One of the two bags is selected at random and a ball is drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first bag.

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906. Bag I contains 3 red and 4 black balls while another Bag II contains 5 red and 6 black balls. One ball is drawn at random from one of the bags and it is found to be red. Find the probability that it was drawn from Bag II.

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Long Answer Type

907. Of the students in a college, it is known that 60% reside in hostel and 40% are day scholars (not residing in hostel). Previous year results report that 30% of all students who residue in hostel attain A grade and 20% of day scholars attain A grade in their annua! examination. At the end of the year, one student is chosen at random from the college and he has an A grade, what is the probability that the student is a hostler?

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Short Answer Type

908. Bag A contains 2 white and 3 red balls and bag B contains 4 white and 5 red balls. One ball is drawn at random from one of the bags and it is found to be red. Find the probability that it was drawn from the bag B.

Let E₁ and E₂ be the events of the ball being drawn from bag A and bag B respectively and E be the event that the drawn ball is red.
Since both the bags are equally likely to be selected i.e., E₁ and E₂ are mutually exclusive and exhaustive events.
$therefore space space space straight P left parenthesis straight E subscript 1 right parenthesis space equals space 1 half comma space space space space straight P left parenthesis straight E subscript 2 right parenthesis space equals space 1 half$
$straight P left parenthesis straight E divided by straight E subscript 1 right parenthesis space equals space Prob. space of space drawing space straight a space red space ball space from space bag space straight A space equals space 3 over 5 straight P left parenthesis straight E divided by straight E subscript 2 right parenthesis space equals space Prob. space of space drawing space straight a space red space ball space from space bag space straight B space equals space 5 over 9 therefore space space space space space straight P left parenthesis straight E subscript 2 vertical line straight E right parenthesis space equals space fraction numerator straight P left parenthesis straight E subscript 2 right parenthesis. space space straight P left parenthesis straight E space vertical line space straight E subscript 2 right parenthesis over denominator straight P left parenthesis straight E subscript 1 right parenthesis. space straight P left parenthesis straight E space vertical line thin space straight E subscript 1 right parenthesis space plus space straight P left parenthesis straight E subscript 2 right parenthesis. space straight P left parenthesis straight E space vertical line thin space straight E subscript 2 right parenthesis end fraction space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator begin display style 1 half end style cross times begin display style 5 over 9 end style over denominator begin display style 1 half end style cross times begin display style 3 over 5 end style plus begin display style 1 half end style cross times begin display style 5 over 9 end style end fraction space equals space fraction numerator begin display style 5 over 9 end style over denominator begin display style 3 over 5 end style plus begin display style 5 over 9 end style end fraction space equals space fraction numerator begin display style 5 over 9 end style over denominator begin display style fraction numerator 27 plus 25 over denominator 45 end fraction end style end fraction equals space fraction numerator begin display style 5 over 9 end style over denominator begin display style 52 over 45 end style end fraction space space space space space space space space equals space 5 over 9 cross times 45 over 52 space equals 25 over 52.$

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Long Answer Type

909. In answering a question on a multiple choice test, a student either knows the answer or guesses. Let

3 over 4

be the probability that he knows the answer and

1 fourth

be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability

1 fourth.

What is the probability that the student knows the answer given that he answered it correctly?

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910. A laboratory blood test is 99% effective in detecting a certain disease when it is in fact, present. However, the test also yields a false positive result for 0.5% of the healthy person tested (i.e. if a healthy person is tested, then, with probability 0.005, the test will imply he has the disease). If 0.1 percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive?

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