Question

Unsolved Problems: 1. A problem in mathematics is given to three students A, B, and C whose chance of solving it are 1/2, 3/4 and 1/4 respectively. What is the probability that i) Problem will be solved? ii) Exactly one of them will solve? 2. Let A and B be two events associated with an experiment. Suppose P (A) = 0.4 while P(A U B) 0.7. Let P(B)p. For what choice of p a) A and B are mutually exclusive b) A and B are independent 3. In the above figure probability of the closing of each relay of the circuit is p. If all the relays function independently, what is the probability that a current exits between terminal L and R? 4. Consider a large lot of items, say 1000. Suppose that 10% of these items are defective and 90% are non defective. Two items are chosen, what is the probability that both the items are non defective? Unsolved Problems: I. In a bolt factory, machines A, B and C manufacture respectively 25%, 35% and 40% of all total. Of their output 5,4, 2 percent are defective bolts. A bolt is drawn at random from the product and is found to be defective. What is the probability that it was manufactured by machines A, B and C. 2.95% of the people who travel by road, use a car, only 5% travel by bus Probability that a journey by car will be completed within schedule time is 0.9, that for a bus the probability 0.2. If a person completes his journey in time. What is the probability that he travelled by car? 3. The content of three urns which contains white, blue and red balls are as below 3R IR 3R One urn is chosen at random and two balls are drawn. They are found to be Urn I W Urn II 2W Urn III 4W 2B 1B 5B white and red. What is the probability that they come from Urn I I or III Lecture notes by Dr. Haseeb Athar, Department of Mathematics, Taibah University, Al-Madinah 4. A table has two drawers each at its left and right corners. Left corner has a gold coin in one drawer and silver coin in the other drawer, while right corner has a gold in each drawer. One corner is picked randomly, then a drawer is chosen at random from the picked corner. The coin in that drawer is found to be gold. What is probability that the coin is from the right corner?

Answer 1

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Answer 2

Answer 1)

chances of solving the problem for A = p(A) = 1/2

chances of solving the problem for B = p(B) = 3/4

chances of solving the problem for C = p(C) = 1/4

i) Probability that the problem will be solved = P(AUBUC)

                                                                      = P(A) + P(B) + P(C) - P(AB) - P(BC) - P(CA) + P(ABC)

                                                                      = 1/2 + 3/4 + 1/4 - ((1/2)*(3/4)) - ((3/4)*(1/4)) - ((1/4)*(1/2)) + ((1/2)*(3/4)*(1/4))

                                                                      = 6/4 - 3/8 - 3/16 - 1/8 + 3/32

                                                                      = (48 - 12 - 6 - 4 + 3) / 32

                                                                      = 29/32 = 0.90625

ii)

p(not A) = 1- p(A) =1 -1/2 = 1/2

p(not B) = 1- p(B) =1 -3/4 = 1/4

p(not C) = 1- p(C) =1 -1/4 = 3/4

Probability exactly one of them will solve = P(A)*P(not B)*P(not C) + P(not A)*P(B)*P(not C)+ P(not A)*P(not B)*P(C)
                                                                 = (1/2)*(1/4)*(3/4) + (1/2)*(3/4)*(3/4) + (1/2)*(1/4)*(1/4)
                                                                 = (3/32) + (9/32) + (1/32)

                                                                 = 13/32 = 0.40625

Answer 2)

P(A) = 0.4

P(B) = p

P(AUB) = 0.7

a)

if A and B are mutually exclusive then

P(AUB) = P(A) + P(B)

0.7 = 0.4 + p

p = 0.7 -0.4 = 0.3

b)

if A and B are independent

P(A ∩ B) = P(A)*P(B)

and P(AUB) = P(A) + P(B) - P(A ∩ B)

0.7 = 0.4 + p - 0.4*p

p = (0.7 - 0.4) / 0.6

p = 0.5