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

b) well because you get well