Ashop receives a shipment of 1000 cheaply made lamps. the probability that any individual lamp is defective is 0.001. assume the defectiveness is independent for each lamp. let x be the number of defective lamps in the batch of 1000. (a) what is the state space of x? (b) what kind of probability mass function does x have? (c) what is the probability that none of the lamps are defective? (d) what is the probability that more than 2 of the lamps are defective?

a) The state space of X is the number of lamps that are defective.

b) Binomial probability mass function.

c) There is a 36.77% probability that none of the lamps are defective.

d) There is a 8.02% probability that more than 2 of the lamps are defective

Step-by-step explanation:

(a) What is the state space of X?

The state space of X is the number of lamps that are defective.

(b) What kind of probability mass function does X have?

X has a binomial probability function mass, because this is the probability of exactly x successes on n repeated trials, and X can only have two outcomes(either the lamp is defective or it is not).

The binomial probability is given by the following formula:

In which is the number of different combinatios of x objects from a set of n elements, given by the following formula.

And p is the probability of a success.

In this problem

There are 1000 total lamps, so .

The probability that any individual lamp is defective is 0.001. A success is a lamp not being defective, so

The probability of there being x defective pieces is given by the following formula

(c) What is the probability that none of the lamps are defective?

This is

There is a 36.77% probability that none of the lamps are defective.

(d) What is the probability that more than 2 of the lamps are defective?

This is . The easier way to calculate this is by subtracting 1 by the probability that there is less than 2 defective lamps, by the following formula:

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So:

There is a 8.02% probability that more than 2 of the lamps are defective

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step-by-step explanation:

not only in mathematics, scientific proof also are required in the other field of science. also maths proofs are of particular structure, containing steps and logics. so proving is like doing science, it is science. that's why it is important to prove. 

an inaccurate proof leads to errors, or stating results that are not accurate.  which wil mess the probelm up you dont want this