Q. A cupboard A has 4 red carpets and 4 blue carpets and a cupboard B has 3 red carpets and 5 blue carpets. A carpet is selected from a cupboard and the carpet is chosen from the selected cupboard such that each carpet in the cupboard is equally likely to be chosen. Cupboards A and B can be selected in \(\frac{1}{5}\) and \(\frac{3}{5}\) ways respectively. Given that a carpet selected in the above process is a blue carpet, find the probability that it came from the cupboard B.

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Q. Mangoes numbered 1 through 18 are placed in a bag for delivery. Two mangoes are drawn out of the bag without replacement. Find the probability such that all the mangoes have even numbers on them?

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Q. A family has two children. Given that one of the children is a girl and that she was born on a Monday, what is the probability that both children are girls?

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Q. Suppose a fair eight-sided die is rolled once. If the value on the die is 1, 3, 5 or 7 the die is rolled a second time. Determine the probability that the sum of values that turn up is at least 8?

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Q. A jar containing 8 marbles of which 4 red and 4 blue marbles are there. Find the probability of getting a red given the first one was red too.

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Q. A bin contains 4 red and 6 blue balls and three balls are drawn at random. Find the probability such that both are of the same color.

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Q. A bucket contains 6 blue, 8 red and 9 black pens. If six pens are drawn one by one without replacement, find the probability of getting all black pens?

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Q. What is the sequence depicted by the generating series 4 + 15x² + 10x³ + 25x⁵ + 16x⁶+⋯?

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Q. What is the generating function for the sequence 1, 6, 16, 216,….?

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Q. What is the generating function for generating series 1, 2, 3, 4, 5,… ?

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Q. What is the generating function for the generating sequence A = 1, 9, 25, 49,…?

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Q. What is the recurrence relation for the sequence 1, 3, 7, 15, 31, 63,…?

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Q. What is multiplication of the sequence 1, 2, 3, 4,… by the sequence 1, 3, 5, 7, 11,….?

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Q. What will be the sequence generated by the generating function 4x/(1-x)²?

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Q. What is the generating function for the sequence with closed formula aₙ=4(7ⁿ)+6(−2)ⁿ?

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Q. Suppose G is the generating function for the sequence 4, 7, 10, 13, 16, 19,…, the find a generating function (in terms of G) for the sequence of differences between terms.

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Q. Find the sequence generated by 1/1−x²−x⁴. (Assume that 1, 1, 2, 3, 5, 8,… has generating function 1/1−x−x².)

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Q. There are 70 patients admitted in a hospital in which 29 are diagnosed with typhoid, 32 with malaria, and 14 with both typhoid and malaria. Find the number of patients diagnosed with typhoid or malaria or both.

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Q. At a software company, skilled workers have been hired for a project. Out of 75 candidates, 48 of them were software engineer; 35 of them were hardware engineer; 42 of them were network engineer; 18 of them had skills in all three jobs and all of them had skills in at least one of these jobs. How many candidates were hired who were skilled in exactly 2 jobs?

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Q. The numbers between 1 and 520, including both, are divisible by 2 or 6 is _______

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