Linear Programming and Probability Quiz

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Questions and Answers

The maximum value of (x + 2y) under the constraints 2x+3y≤6, x + 4y ≤ 4, x, y ≥0 is

  • 3.2 (correct)
  • 3
  • 2
  • 4

The incidence of occupational disease in an industry is such that the workmen have a 10% chance of suffering from it. The probability that out of 5 workmen, 3 or more will contract the disease is

  • 0.0856 (correct)
  • 0.0000856
  • 0.00856
  • 0.000856

In binomial distribution if n = 25 E (X) = 10, then var (X) =

2.5

If a fair coin is tossed 8 times, then the probability that it shows heads at least once is

<p>255/256 (D)</p> Signup and view all the answers

The probability that a person who undergoes a kidney operation will recover is 0.7. If the six patients who undergoes similar operations, then the probability that half of them will recover is

<p>0.3704 (D)</p> Signup and view all the answers

In a test of Mathematics, there are two types of questions to be answered-short answered and long answered. The relevant data is given below. Time taken to solve: Short answered questions 5 minute; Long answered questions 10 minute. Marks: Short answered questions 3; Long answered questions 5. Number of questions: Short answered questions 10; Long answered questions 14. The total marks is 100. Students can solve all the questions. To secure maximum marks, a student solves x short answered and y long answered questions in three hours, the linear constraints are x≥ 0, y ≥ 0, x ≤10 , y ≤14 and 5x+10y≤180. Then the vertices of a feasible region are

<p>(0, 18), (10, 13) (D)</p> Signup and view all the answers

The position of points O (0, 0) and P (2, - 2) in the region of graph of inequation 2x – 3y <5, will be

<p>O outside and P inside (C)</p> Signup and view all the answers

Find out maximum value of z=5x+7y subject to x + y ≤ 4, 3x + 8y ≤ 24, 10x + 7y ≤ 35, x, y ≥ 0.

<p>42.8</p> Signup and view all the answers

A vertex of the linear inequalities 2x+3y≤6, x+4y≤4 and x, y ≥ 0, is

<p>(1, 1) (C)</p> Signup and view all the answers

In a box containing 100 eggs, 10 eggs are rotten. The probability that out of a sample of 5 eggs none are rotten, if the sampling is with replacement, is

<p>(9/10)^5 (C)</p> Signup and view all the answers

In binomial probability distribution, mean is 3 and standard deviation is 3/2. Then the probability distribution is

<p>3/4 + 3/4 (B)</p> Signup and view all the answers

The corner points of the feasible region determined by the system of linear constraints are (0,10), (5,5) (15,15), (0,20). Let z = px + qy, where p, q > 0. Condition on p and q so that the maximum of z occurs at both the points (15, 15) and (0, 20) is

<p>q = 3p (A)</p> Signup and view all the answers

If a fair coin is tossed 8 times, then the probability that it shows heads exactly 5 times is

<p>7/32 (A)</p> Signup and view all the answers

If for a binomial distribution X~B (7, p) P (X = 1) = P ( X = 2), then p is

<p>1/4 (D)</p> Signup and view all the answers

An urn contains 4 white and 3 red balls. If 3 balls ar are drawn one by one with replacement and probability of getting exactly two red ball is a(3/b)³, then a+b is

<p>11</p> Signup and view all the answers

The points which provides the solution to the linear programming problem: Max P = 2x + 3y subject to constraints: x≥ 0, y ≥ 0, 2x + 2y ≤ 9, 2x+y≤7,x+2y≤8 is

<p>(2, 3.5) (C)</p> Signup and view all the answers

Two different kinds of food A and B are being considered to form a weekly diet. The minimum weekly requirement for fats, carbohydrates and proteins are 18, 24 and 16 units respectively. One kg of food A has 4, 16 and 8 units respectively of these ingredients and one kg of food B has 12, 4 and 6 units respectively. The price of food A is Rs.4 per kg and that of food B is Rs.3 per kg. Find out minimum cost.

<p>35/22 units of food A and 41/11 units of food B, minimum cost is Rs. 90 (C)</p> Signup and view all the answers

If p.m.f. of r.v. X is P(X)=4(4-x)/10 , x = 0,1,2,3,4, then Var(X)=

<p>0.9786</p> Signup and view all the answers

Inequations 3x-y≥3 and 4x-y> 4

<p>have solution for positive x and y (A)</p> Signup and view all the answers

The values of x and y for which the objection function z = 3x + 4y under the constraints y ≤x+2,4x ≤ x + 2, 4x +3 + 3y ≤ 12, x > 0 , y ≥ 0 is maximum are

<p>x=0 and y=2 (D)</p> Signup and view all the answers

Shaded region is represented by

<p>2x + 5y ≤ 80, x + y ≤ 20, x ≥ 0, y ≥ 0 (D)</p> Signup and view all the answers

An insurance agent insures lives of 5 men, all having same age and good health. The probability that a man of this age will survive the next 30 years is known to be 2/3. What is the probability that in the next 30 years almost 3 men will survive?

<p>163/243 (B)</p> Signup and view all the answers

One coin is thrown 100 times, then the probability of getting head in odd number is

<p>3/8 (D)</p> Signup and view all the answers

An experiment succeeds twice as often as it fails. Find the probability that in 4 trials there will be at least three success.

<p>23/24 (C)</p> Signup and view all the answers

For an L.P.P. the feasible region is shown shaded in the figure. Find the maximum value of the objective function z = 5x +7y

<p>25</p> Signup and view all the answers

The point at which, the maximum value of (3x + 2y) subject to the constraints x+y≤2, x≥0, y ≥0 obtained, is

<p>(2, 0) (B)</p> Signup and view all the answers

For a bionomial distribution X~B (n, p) if n=5 and P(X = 1) = 8P (X = 3), then mean of the distribution is

<p>1 (D)</p> Signup and view all the answers

The minimum value of objective function c = 2x + 2y in the given feasible region, is

<p>38 (A)</p> Signup and view all the answers

A die is thrown 6 times. If "getting an odd number" is a success, then the probability of getting 5 successes is....

<p>3/32 (C)</p> Signup and view all the answers

Flashcards

Linear Inequalities

Equations involving inequalities instead of equality, defining a region on a graph.

Feasible Region

Set of all possible points that satisfy the given constraints.

Maximize Objective Function

Finding the highest value of a function under given constraints.

Vertices of Feasible Region

Corner points of the feasible region where maximum/minimum values occur.

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Probability of Event

The likelihood of an event occurring, calculated as favorable outcomes over total outcomes.

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Binomial Distribution

Probability distribution of a number of successes in a fixed number of trials.

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Expected Value (E(X))

The long-term average value of a random variable.

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Variance

A measure of how far each number in the set is from the mean, telling variability.

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Standard Deviation

The square root of variance, indicating spread in data and consistency.

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Combinations

The selection of items from a larger pool where order does not matter.

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Constraints

Restrictions or limits imposed on variables in a mathematical model.

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P(more than 3 successes)

Probability of obtaining more than three successful outcomes.

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Inequation

An inequality that describes a relationship between expressions.

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Sampling with Replacement

Selecting items from a population where each item can be chosen more than once.

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Sampling without Replacement

Selecting items from a population where once selected, the item cannot be chosen again.

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Binomial Probability Formula

Formula used to calculate the probability of a given number of successes in a binomial setting.

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Objective Function in LPP

A function that needs to be maximized or minimized in a linear programming problem.

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Probability Mass Function (PMF)

Function that gives the probability of discrete outcomes.

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Conditional Probability

The probability of an event given that another event has occurred.

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Multiple Events Probability

Calculating the probability involving more than one event.

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Shaded Region

The area on a graph representing solutions that satisfy inequalities.

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Linear Programming Problem (LPP)

A problem where a linear objective function is maximized or minimized subject to constraints.

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Random Variable

A numerical outcome of a random process.

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Probabilistic Events

Events whose outcomes can be described by probability.

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Critical Values

Values at which the value of a function or a dependent variable changes.

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Study Notes

Questions and Answers

  • Question 1: Find the maximum value of (x + 2y) given constraints 2x + 3y ≤ 6, x + 4y ≤ 4, x, y ≥ 0
  • Question 2: Find the probability that 3 or more out of 5 workmen in an industry will contract an occupational disease given a 10% chance of contracting it.
  • Question 3: If n = 25 and E(X) = 10 in a binomial distribution, find Var(X).
  • Question 4: Find the probability of getting at least one head in 8 coin tosses.
  • Question 5: A kidney operation recovery rate is 0.7. What is the probability that half of 6 patients recover?
  • Question 6: A math test has short-answer and long-answer questions with respective time and mark allocations. What is the optimal strategy to maximize marks?
  • Question 7: Given points O(0, 0) and P(2, -2), determine if they are inside or outside the region of the inequality 2x - 3y < 5.
  • Question 8: Find the maximum value of z = 5x + 7y subject to x + y ≤ 4, 3x + 8y ≤ 24, 10x + 7y ≤ 35, x, y ≥ 0.
  • Question 9: Find a vertex of the following inequalities: 2x + 3y ≤ 6, x + 4y ≤ 4, x, y ≥ 0.
  • Question 10: 100 eggs in a box, 10 rotten. Probability that a sample of 5 eggs contains none rotten - with replacement sampling.
  • Question 11: Binomial probability distribution, mean = 3, standard deviation = 3/2. Find the probability distribution.
  • Question 12: Find the minimum cost for acquiring units of food A, priced at Rs.4 per kg, and units of food B, priced at Rs.3 per kg.
  • Question 13: An event has heads occurring exactly 5 times in 8 coin tosses. Find the probability.
  • Question 14: Solve for p in a binomial distribution given the conditions X ~ B(7, p), P(X = 1) = P(X = 2).
  • Question 15: A container has 4 white and 3 red balls. Find the probability that, with replacement, drawing 3 balls yields exactly 2 red balls.
  • Question 16: Find the solution points for a linear programming problem with given constraints.
  • Question 17: A diet problem involving two types of food A and B, considering weekly minimum requirements of fats, carbohydrates, and proteins. Find weekly diet
  • Question 18: A random variable X with given probability mass function (PMF): P(X) = (4-x)/5 with x = 0, 1, 2, 3, 4. Find Var(X).
  • Question 19: Determine if the system of inequalities 3x-y ≥ 3 and 4x - y > 4 have solutions for all x, y, or for specific positive values.
  • Question 20: Find the values of x and y for which an objective function z = 3x + 4y is maximized under specific constraints.
  • Question 21: Identify the shaded region represented in the graph.
  • Question 22: Find the probability that exactly 3 men out of 5 will survive 30 years.
  • Question 23: Throwing a coin 100 times, find the probability of getting heads in an odd number of times.
  • Question 24: In an experiment, success occurs twice as often as failure. Find the probability of at least 3 successes in 4 trials.
  • Question 25: A question on finding the maximum value of an objective function (z = 5x + 7y) in a linear programming problem with given constraints.
  • Question 26: Find the point that maximizes (3x + 2y) under given constraints in a linear programming problem.
  • Question 27: A binomial distribution question with n = 5 trials. Find the mean of binomial distribution with given P(X = 1) and P(X = 3)
  • Question 28: Find the minimum value of objective function c = 2x + 2y in a given feasible region.
  • Question 29: A die is thrown 6 times. Find P(getting an odd number 5 times)
  • Question 30: Find the common solution for constraints 2x + 3y ≤ 134, x + 5y ≤ 200, x ≥ 0, y ≥ 0.

Additional Topics (from other pages)

  • Linear programming: Maximizing objective functions under constraints, feasible regions, corner points.
  • Binomial distribution: Mean, variance, probabilities, conditions.
  • Probability: Calculations involving probabilities of events.
  • Statistics: Concepts and applications of basic statistical measures
  • Inequalities: Solving systems of inequalities.

Additional Notes (from other pages)

  • Problem Solving Strategies: Understanding how to approach various problems with given constraints, objective functions, and probabilities
  • Mathematical Models: Linear programming application to real-world scenarios, involving resources, costs, profits.

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