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

#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#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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#