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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.

Bag I, Bag II
Number of white balls = 3, Number of white balls = 5
Number of red balls = 2, Number of red balls = 4
Total number of balls = 5, Total number of balls = 9
Let E₁ and E₂ be the events of the ball being drawn from bag I and bag II respectively and E be the event that the drawn ball is red.
Since both the bags are equally likely to be selected
$therefore space space space space straight P left parenthesis straight E subscript 1 right parenthesis space equals space 1 half comma space space straight P left parenthesis straight E subscript 2 right parenthesis space equals space 1 half$
$straight P left parenthesis straight E left enclose straight E subscript 1 end enclose right parenthesis space equals space Probability space of space drawing space straight a space red space ball space from space bag space straight I space equals space 2 over 5 straight P left parenthesis straight E space left enclose straight E subscript 2 end enclose right parenthesis space equals space Probability space of space drawing space straight a space red space ball space from space bag space II space equals space 4 over 9$
P(the red ball is drawn from bag II) = $straight P left parenthesis straight E subscript 2 space end subscript left enclose straight E right parenthesis$
$equals space fraction numerator straight P left parenthesis straight E subscript 2 right parenthesis. space space straight P left parenthesis straight E subscript 1 left enclose straight E subscript 2 end enclose right parenthesis over denominator straight P left parenthesis straight E subscript 1 right parenthesis. space straight P left parenthesis straight E left enclose straight E subscript 1 end enclose right parenthesis space plus space straight P left parenthesis straight E subscript 2 right parenthesis. space space straight P left parenthesis straight E left enclose straight E subscript 2 end enclose right parenthesis end fraction equals space fraction numerator begin display style 1 half end style cross times begin display style 4 over 9 end style over denominator begin display style 1 half end style cross times begin display style 2 over 5 end style plus begin display style 1 half end style cross times begin display style 4 over 9 end style end fraction space equals fraction numerator begin display style 2 over 9 end style over denominator begin display style 1 fifth end style plus begin display style 2 over 9 end style end fraction space equals fraction numerator begin display style 2 over 9 end style over denominator begin display style fraction numerator 9 plus 10 over denominator 45 end fraction end style space end fraction equals space fraction numerator begin display style 2 over 9 end style over denominator begin display style 19 over 45 end style end fraction equals space 2 over 9 cross times 45 over 19 space equals space 10 over 19$

83 Views

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.

74 Views

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.

75 Views

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.

158 Views

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.

83 Views

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?

98 Views

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?

93 Views

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?

90 Views