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# Probability- lifetime of light bulbs

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A certain factory has three machines A,B and C that produces one type of light bulbs.Past experience has shown that the lifetime of a light bulb by machine A can be modeled as an exponential random variable with an average of 20 days, whereas the lifetime of a bulb produced by a machine B can modeled as a normal random variable with mean of 15 days and variance of 5.76(days^2).As for the light bulbs produced by machine C, they usually last anywhere between 13 to 17 days.It is further known that 30% of the factory production of this type of light bulbs comes from machine A and 50% of production comes from machine B.Suppose all the light bulbs after production are packaged identically and stored in the same storage room.
a)A bulb is selected at random from the storage room and tested.Find the probability that the bulb was produced by machine C given that it was still operating after 16 days.
b) A random sample of size 10 bulbs is selected from the storage room, what is the probability that exactly 4 of the bulbs will have a lifetime less than 16 days?
c)on average ,how many bulbs must be selected in order to get 7 bulbs with a lifetime between two and three weeks?
d)Find the expacted lifetime of a lifetime of a light bulb produced by this factory.Hint:use the Law Of Total Expectation.

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The expert calculates the probabilities dealing with lifetime of light bulbs.

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A certain factory has three machines A,B and C that produces one type of light bulbs.Past experience has shown that the lifetime of a light bulb by machine A can be modeled as an exponential random variable with an average of 20 days, whereas the lifetime of a bulb produced by a machine B can modeled as a normal random variable with mean of 15 days and variance of 5.76(days^2).As for the light bulbs produced by machine C, they usually last anywhere between 13 to 17 days.It is further known that 30% of the factory production of this type of light bulbs comes from machine A and 50% of production comes from machine B.Suppose all the light bulbs after production are packaged identically and stored in the same storage room.
a)A bulb is selected at random from the storage room and tested.Find the probability that the bulb was produced by machine C given that it was still operating after 16 days.
b) A random sample of size 10 bulbs is selected from the storage room, what is the probability that exactly 4 of the bulbs will have a lifetime less than 16 days?
c)on average ,how many bulbs must be selected in order to get 7 bulbs with a lifetime between two and three weeks?
d)Find the expacted lifetime of a lifetime of a light bulb produced by this factory.Hint:use the Law Of Total Expectation

a)A bulb is selected at random from the storage room and tested.Find the probability that the bulb was produced by machine C given that it was still operating after 16 days.
Machine A:
machine A can be modeled as an exponential random variable with an average of 20 days

The exponential random variable is represented as :
P(x)= lambda x e -lambda / x!
where
P(x)= probability of x
lambda = average

To calculate the probability that the bulb is operating after 16 days we need to calculate the probabilities of failure in 0,1,2,3---16 days
Here lambda = ...

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