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Probability

Long Answer Type

911. Suppose that 5% of men and 0.25% of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male? Assume that there are equal number of males and females.

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Short Answer Type

912. If a machine is correctly set up, it produces 90% acceptable items. If it is incorrectly set up, it produces only 40% acceptable items. Past experience shows that 80% of the set ups are correctly done. If after a certain set up, the machine produces 2 acceptable items, find the probability that the machine is correctly setup.

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Long Answer Type

913. Assume that the chances of a patient having a heart attack are 40%. It is also assumed that a meditation and yoga course reduce the risk of heart attack by 30% and prescription of a certain drug reduces its chance by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?

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Short Answer Type

914. A factory has two machines A and B. Past record shows that machine A produced 60% of the items of output and machine B produced 40% of the items. Further, 2% of the items produced by machine A and 1% produced by machine B were defective. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that it was produced by machine B?

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915. Two groups are competing for the position on the Board of directors of a corporation: The probabilities that the first and the second groups will win are 0.6 and 0.4 respectively. Further, if the first group wins, the probability of introducing a new product is 0.7 and the corresponding probability is 0.3 if the second group wins. Find the probability that the new product introduced was by the second group.

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Long Answer Type

916. Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets 1, 2, 3 or 4. she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw 1, 2, 3 or 4 with the die?

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917. A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond.

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918.
Suppose that the reliability of HIV test is specified as follows:
Of people having HIV, 90% of the test detect the disease but 10% go undetected. Of people free of HIV, 99% of the test are judged HIV -ve but 1% are diagonsed as showing HIV +ve. From a large population of which only 0.1% have HIV, one person is selected at random, given the HIV test, and the pathologist reports him/her as HIV +ve. What is the probability that the person actually has HIV?

Let E denote the event that the person selected is actually having HIV and A the event that the person's HIV test is diagnosed as +ve.
Now

straight P left parenthesis straight E right parenthesis space equals space 0.1 percent sign space equals space fraction numerator 0.1 over denominator 100 end fraction space equals space 0.001

straight P left parenthesis straight E apostrophe right parenthesis space equals space 1 minus space straight P left parenthesis straight E right parenthesis space equals space 0.999

straight P left parenthesis straight A space vertical line thin space straight E right parenthesis space equals space straight P left parenthesis person space tested space as space HIV space plus space ve space given space that space he divided by she space is space actually space having space HIV right parenthesis

equals space 90 percent sign space equals space 90 over 100 space equals 0.9

and P(A | E') = P(person tested as HIV +ve given that he/she is actually not having HIV}

equals space 1 percent sign space equals space 1 over 100 space equals space 0.01

Now, by Baye's theorem

straight P left parenthesis straight E space vertical line space straight A right parenthesis space equals space fraction numerator straight P left parenthesis straight E right parenthesis space straight P left parenthesis straight A space left enclose straight E right parenthesis over denominator straight P left parenthesis straight E right parenthesis space straight P left parenthesis straight A space left enclose straight E right parenthesis space plus space straight P left parenthesis straight E apostrophe right parenthesis space straight P left parenthesis straight A space left enclose straight E apostrophe right parenthesis end fraction space space space space space space space space space space space space space space space equals space fraction numerator 0.001 space cross times 0.9 over denominator 0.001 cross times 0.9 plus 0.999 cross times 0.01 end fraction equals space fraction numerator 0.0009 over denominator 0.009 plus 0.00999 end fraction equals fraction numerator 0.0009 over denominator 0.01089 end fraction space space space space space space space space space space space space space space space space equals space fraction numerator begin display style 9 over 10000 end style over denominator begin display style 1089 over 100000 end style end fraction space equals 90 over 1089 space equals 0.83 space nearly

∴ the probability that a person selected at random is actually having HIV given that he/she is tested HIV +ve is 0·083.

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Short Answer Type

919. A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.

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920. There are three coins. One is a two headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin?

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