a) What is the probability that the entire shipment shall be accepted?
b) What is the probability that the entire shipment shall be rejected?

Question

A shipment of 40 Mickey Mouse watches contains 6 defective ones. The shipping department selects seven of these watches and rejects the entire shipment if one or more are defective.

Ethan Adkins · Accepted Answer

a) All 7 meet specification #->q^7 = (17/20)^7 ~~0.3206 # to 4 dp

b) At least 1 fail (rejected) #1-q^7 =1-(17/20)^7~~0.6794 # to 4dp

#color(green)("Standard notation for quality assurance gives:")#

#color(green)("Let probability of good be "q)#
#color(green)("Let probability of defective be "p " think "p->"poor quality'"#

The total ways of ordering this for a sample of 7 is: #(p+q)^7#

Using Pascal's triangle for Binomial expansion we select #x^7#:

#x^7" "->" "1+7+21+35+35+21+7+1#

#1p^7q^0+7p^6q^1+21p^5q^2+35p^4q^3+35p^3q^4+21p^2q^5+7p^1q^6+1p^0q^7#

#p^7+7p^6q^1+21p^5q^2+35p^4q^3+35p^3q^4+21p^2q^5+7p^1q^6+q^7#
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Accept #-># condition that has all good which is the last one of #q^7#
Reject #-># condition NOT all good which is #1-q^7#

Defective probability given as:#" "p=6/40 = 3/20#
Not defective is thus: #" "q=1-p=1-3/20=17/20#

a) All 7 meet specification #->q^7 = (17/20)^7 ~~0.3206 # to 4 dp

b) At least 1 fail (rejected) #1-q^7 =1-(17/20)^7~~0.6794 # to 4dp
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Comment.

It is more likely the shipment will be rejected. This is very costly for the supplier. So it would be far better for them to apply 100% inspection at the manufacture point.

The advantage of using the Binomial expansion is that it permits the probability calculation for any of the individual condition (or groups).

Suppose I wished to determine 3 poor and 4 quality. I would use the #35p^3q^4#

#=35xx(3/20)^3xx(17/20)^4#

Jameson Allen · Answer

undefined To determine the probability of the entire shipment being accepted or rejected, we need to know the individual probabilities of each item being accepted or rejected and then use these probabilities to calculate the overall probability.

Let's denote:
- $p$ as the probability that an item is accepted.
- $q$ as the probability that an item is rejected.

a) To find the probability that the entire shipment shall be accepted, we multiply the probability of each item being accepted together since the acceptance of each item is independent:

$$ P(\text{Entire shipment accepted}) = p^n $$

where $ n $ is the number of items in the shipment.

b) Similarly, to find the probability that the entire shipment shall be rejected, we multiply the probability of each item being rejected together:

$$ P(\text{Entire shipment rejected}) = q^n $$

It's important to note that $ q = 1 - p $, where $ p $ is the probability of acceptance.

a) What is the probability that the entire shipment shall be accepted? b) What is the probability that the entire shipment shall be rejected?

A shipment of 40 Mickey Mouse watches contains 6 defective ones. The shipping department selects seven of these watches and rejects the entire shipment if one or more are defective.