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You take an exam that contains 20 multiple-choice questions. Each question has 4 possible options. You know the answer to 10 questions, but you have no idea about the other 10 questions so you choose answers randomly. Your score X on the exam is the total number of correct answers. Find the p.m.f. of X. What is P(X > 15)?
I roll a fair die repeatedly until a number larger than 4 is observed. If N is the total number of times that I roll the die, find P(N = k), for k = 1,2,3,....
Find the variance of the sampling distribution of means 2,5,7,8.
Each of the 50 students in class belongs to exactly one of the four groups A, B, C, or D. The membership numbers for the four groups are as follows: A: 5, B: 10, C: 15, D: 20. First, choose one of the 50 students at random and let X be the size of that student’s group. Next, choose one of the four groups at random and let Y be its size. (Recall: all random choices are with equal probability, unless otherwise specified.)
(a) Write down the probability mass functions for X and Y . (b) Compute E[X] and E[Y ] . (c) Compute V (X) and V (Y ). (d) Assume you have s students divided into n groups with membership numbers s1,s2,··· ,sn and again X is the size of the group of a randomly chosen student, while Y is the size of the randomly chosen group. Let E[Y ] = µ and V (Y ) = σ2. Express E[X] with s,n,µ, and σ.
You and your opponent both roll a fair die. If you both roll the same number, the game is repeated, otherwise whoever rolls the larger number wins. Let N be the number of times the two dice have to be rolled before the game is decided.
(a) Determine the probability mass function of N.
(b) Compute E[N].
(c) Compute P(you win).
(d) Assume that you get paid $10 for winning in the first round, $1 for winning in any other round, and nothing otherwise. Compute your expected winnings.
You are dealt one card at random from a full deck and your opponent is dealt 2 cards (without any replacement). If you get an Ace, he pays you $10, if you get a King, he pays you $5 (regardless of his cards). If you have neither an Ace nor a King, but your card is red and your opponent has no red cards, he pays you $1. In all other cases you pay him $1. Determine your expected earnings. Are they positive?
A biologist needs at least 3 mature specimens of a certain plant. The plant needs a year to reach maturity; once a seed is planted, any plant will survive for the year with probability 1/1000 (independently of other plants). The biologist plants 3000 seeds. A year is deemed a success if three or more plants from these seeds reach maturity.
(a) Write down the exact expression for the probability that the biologist will indeed end up with at least 3 mature plants.
(b) Write down a relevant approximate expression for the probability from (a). Justify briefly the approximation.
(c) The biologist plans to do this year after year. What is the approximate probability that he has at least 2 successes in 10 years?
(d) Devise a method to determine the number of seeds the biologist should plant in order to get at least 3 mature plants in a year with probability at least 0:999. (Your method will probably require a lengthy calculation (do not try to carry it out with pen and paper.)
Roll a fair die repeatedly. Let X be the number of 6’s in the first 10 rolls and let Y the number of rolls needed to obtain a 3.
(a) Write down the probability mass function of X.
(b) Write down the probability mass function of Y .
(c) Find an expression for P(X ≥ 6).
(d) Find an expression for P(Y > 10).
Given the Standard Normal distribution, find the following
(a) P(Z < 1.8)
(b) P(−1.1 < Z ≤ 1.8)
(c) P(−1.8 ≤ Z ≤−1.1)
(d) P(Z > −2.5)
(e) P(Z > −0.95)
(f) P(Z < −0.95)
(g) P(Z ≥ 2.18)
(h) P(Z > 10)
The number of claims per month paid by an insurance company is modelled by a random variable N with p.m.f satisfying the relation
p(n + 1) =
1/3p(n), n = 0,1,2,...
where p(n) is the probability that n claims are filed during a given month
(a) Find p(0).
(b) Calculate the probability of at least one claim during a particular month given that there have been at most four claims during the month.
