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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.

Bag I contains 3 red and 4 black balls.
Bag II contain 4 red and 5 black balls.
Let E₁ : Event that a red ball is drawn from bag I
E₂: Event that a black ball is drawn from bag I
$therefore space space space space space straight P left parenthesis straight E subscript 1 right parenthesis space equals space 3 over 7 comma space space straight P left parenthesis straight E subscript 2 right parenthesis space equals space 4 over 7$

After transferring a red ball from bag I to bag II, the bag II will have 5 red and 5 black balls.
Let A be the event of drawing red ball
$therefore space space space space straight P left parenthesis straight A space left enclose straight E subscript 1 end enclose right parenthesis space equals space 5 over 10 space equals 1 half$
Further when a black ball is transferred from bag I to bag II, it will contain 4 red and 6 black balls.
$straight P space open parentheses straight A space left enclose straight E subscript 2 end enclose close parentheses space equals space 4 over 10 space equals space 2 over 5$
Required probability = $straight P left parenthesis straight E subscript 2 space left enclose space straight A end enclose right parenthesis space equals space fraction numerator straight P left parenthesis straight E subscript 2 right parenthesis thin space straight P left parenthesis straight A space left enclose space straight E subscript 2 end enclose right parenthesis over denominator straight P left parenthesis straight E subscript 1 right parenthesis space straight P left parenthesis straight A space left enclose straight E subscript 1 end enclose right parenthesis space plus space straight P left parenthesis straight E subscript 2 right parenthesis thin space straight P left parenthesis straight A space left enclose straight E subscript 2 end enclose right parenthesis end fraction space equals space fraction numerator begin display style 4 over 7 end style cross times begin display style 2 over 5 end style over denominator begin display style 3 over 7 end style cross times begin display style 1 half end style plus begin display style 4 over 7 end style cross times begin display style 2 over 5 end style end fraction space equals space fraction numerator begin display style 8 over 35 end style over denominator begin display style 13 over 14 end style plus begin display style 8 over 35 end style end fraction space equals space fraction numerator begin display style 8 over 35 end style over denominator begin display style fraction numerator 15 plus 16 over denominator 70 end fraction end style end fraction space equals 8 over 35 cross times 70 over 31 equals 16 over 31.$

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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.

85 Views

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.

72 Views

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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