Question 1 Suppose that a cable is composed of a number of i

Question 1 Suppose that a cable is composed of a number of independent wires. Occasionally the cable is subject to high overloads, on such occasions the probability that one wire will fracture is 0.05. Assume that the failure of 2 or more wires during an overload is unlikely. If the cable must be replaced when 3 of the wires have failed, determine the probability that the cable can withstand at least 5 overload applications before being replaced. Round to five significant figures CX xxxxox). Question 2 There exists a 40 foot soil stratum where boulders are randomly deposited. Piles are designed to be driven into the rock. For simplicity, assume that the stratum can be divided into 4 independent layers of 10 ft. each. The probability of hitting a boulder within each 10-ft layer is 0.02, and that the probability of hitting 2 or more boulders within each layer is negligible. What is the probability that a pile will be successfully driven into the rock without hitting any boulder? Question 3 Refer to Q2.What is the probability that it will hit at most 1 boulder on its way to rock? Question 4 Refer to Q2. What is the probability that a pile will hit the first boulder in the third layer?

Solution

Possion Distribution
PMF of P.D is = f ( k ) = e-k ?x / x!
Where   
k = parameter of the distribution.
x = is the number of independent trials

Probability of hitting a boulder in 4 layers will be 0.02*4 = 0.08
Q2.
P( X = 0 ) = e ^-0.08 * 0.08^0 / 0! = 0.9231

Q3.
P( X < = 1) = P(X=1) + P(X=0) +   
= e^-0.08 * 0.08 ^ 1 / 1! + e^-0.08 * 0 ^ 0 / 0! +
= 0.997

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