Answer:
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