Question

Suppose there are 4 defective items in a lot (collection) of 60 items. A sample of size 15 is taken at random and without replacement. Let X denote the number of defective items in the sample.
please show work..

Answer 1

Given X denote the number of defective items in the sample with probability p = 4/60 = 0.06667 and the sample size is 15.

So the distribution X ~ B(15,0.06667)

See the image for the probability table(left) and probability mass function(right).

III Binomial Distribution desmos Create Account ОГ Sign In << 11 0 + -0.6 4 Х B(n.p,x) 0 0.35526436 1 0.38064039 2 0.1903202 0.4 3 0.058908633 0.012623279 5 0.0019836581 6 2.361498 x 10-4 7 2.168722 x 10-5 0.2 8 1.549087 x 10-6 9 8.606041 x 10-8 10 3.688303 x 10-9 11 1.197501 × 10-10 -2 0 2 4 6 8 10 12 14 16 18 20 22 12 2.851193 x 10-12 13 4.699769 10-14 14 4.795683 x 10-16 15 2.283658 x 10-18 --0.2 Х

Now in last part, it is asked P(X<=2 | X > 1)

which is P(X<= 2  $\bigcap$ X>1 ) / P(X>1) ==> P(X=2) /1 - P(X<=1)

P(X=2) = 0.190 and P(X<=1) = 0.355+0.380 = 0.735 (Note: The values are taken from probability table in image)

==> P(X=2) /1 - P(X<=1) = 0.190/ 1-0.735 =  0.7169