Question

In a shipment of 10,000 of a certain type of the electronic chips, 300 chips are defective. The manufacturer takes a random sample of 50 chips without replacement and examines them. If there are at least three defective chips, the manufacturer replaces the shipment

(a) What is the propbability that the manufacturer does not need to replace the shipment? 2 (b) What is the approximate probability that the manufacturer does not need to replace the shipment? (c) Can you approximate the probability found in part(b) further?

Answer 1

Solution :

In a shipment of 10,000 of a certain type of the electronic chips, 300 chips are defective.

Let we define; X = number of defective in a shipment

Success = getting defective in a shipment

Hence;  

The manufacturer takes a random sample of 50 chips without replacement, that is, n = 50.

And population size; N = 10000.

So

Note that the random sample is taken without replacement but since

So we can assume that each trial of random sample of size 50 is independent. Also P(success ) is same for each of the 50 trials.

This fulfil all the conditions of binomial experiment.

And hence we have; X ~ Binomial ( n = 50, p = 0.03 ).

The probability mass function of a binomial random variable is given by;

Note that; if there are at least three defective chips, the manufacturer replaces the shipment, that is, if (X >= 3) then there is replacement of shipment.

( a ).

The probability that the manufacturer does not need to replace the shipment is given by;

The probability that the manufacturer does not need to replace the shipment is 0.8108.

( b ).

The probability that the manufacturer does not need to replace the shipment is given by;

We can use normal approximation to binomial distribution to find this probability.

That the binomial random variable X is approximated to a normal random variable Y with,

While using normal approximation to binomial random variable X we use continuity correction factor as follows;

The approximated probability that the manufacturer does not need to replace the shipment is ;

0.7964.

( c ).

We can not approximate the probability obtained in part ( b ) further.