There are 40 items in which 8 are defective. if 6 items are selected at random, what is the probability that 4 are non defective?
To find the probability that 4 out of 6 selected items are non-defective, we first need to calculate the total number of possible outcomes and the number of favorable outcomes.
Total number of possible outcomes:
Since we are selecting 6 items out of a total of 40, we can use the combination formula. The formula for combinations is nCr, where n is the total number of items and r is the number of items selected. In this case, n = 40 and r = 6.
nCr = n! / (r!(n-r)!)
40C6 = 40! / (6!(40-6)!)
= 40! / (6!34!)
Favorable outcomes:
To have 4 non-defective items, we need to choose 4 non-defective items from the 32 non-defective items available, and choose 2 defective items from the 8 defective items available. We can again use the combination formula to calculate the number of favorable outcomes.
Favorable outcomes = (32C4) * (8C2)
= (32! / (4!(32-4)!) ) * (8! / (2!(8-2)!))
= (32! / (4!28!)) * (8! / (2!6!))
= (32! / (4!28!)) * (8! / (2!6!))
Now we can calculate the probability:
Probability = Favorable outcomes / Total outcomes
= [(32! / (4!28!)) * (8! / (2!6!))] / [40! / (6!34!)]
By calculating this expression, we can find the probability that 4 out of 6 selected items are non-defective.