Q. A Random Variable X can take only two values, 4 and 5 such that P(4) = 0.32 and P(5) = 0.47. Determine the Variance of X.

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Q. A 6-sided die is biased. Now, the numbers one to four are equally likely to happen, but five and six is thrice as likely to land face up as each of the other numbers. If X is the number shown on the uppermost face, determine the expected value of X when 6 is shown on the uppermost face.

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Q. A fair cubical die is thrown twice and their scores summed up. If the sum of the scores of upper side faces by throwing two times a die is an event. Find the Expected Value of that event.

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Q. A random variable X can take only two values, 2 and 4 i.e., P(2) = 0.45 and P(4) = 0.97. What is the Expected value of X?

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Q. A single card is drawn from a standard deck of playing cards. What is the probability that the card is a face card provided that a queen is drawn from the deck of cards?

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Q. Naina receives emails that consists of 18% spam of those emails. The spam filter is 93% reliable i.e., 93% of the mails it marks as spam are actually a spam and 93% of spam mails are correctly labelled as spam. If a mail marked spam by her spam filter, determine the probability that it is really spam.

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Q. A meeting has 12 employees. Given that 8 of the employees is a woman, find the probability that all the employees are women?

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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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