in a shipment of 7 items there are three defective items. at random 4 items are selected what is the expected number of defective items?
To find the expected number of defective items in a shipment of 7 items, where 3 items are defective, and 4 items are randomly selected, we can use the concept of probability.
First, let's calculate the probability of selecting a defective item for each of the 4 selections. We have 3 defective items out of 7 in total:
First selection:
Probability of selecting a defective item = (Number of defective items / Total number of items) = 3/7
Second selection:
Since one defective item has already been selected, there are now 2 defective items remaining, and the total number of items is reduced to 6:
Probability of selecting a defective item in the second selection = 2/6 = 1/3
Third selection:
There are now 2 defective items remaining, and the total number of items remaining is 5:
Probability of selecting a defective item in the third selection = 2/5
Fourth selection:
There is still 1 defective item remaining, and the total number of items remaining is 4:
Probability of selecting a defective item in the fourth selection = 1/4
The expected number of defective items can be calculated by multiplying the probability of selecting a defective item in each selection:
Expected number of defective items = (Probability of first selection) × (Probability of second selection) × (Probability of third selection) × (Probability of fourth selection)
Expected number of defective items = (3/7) × (1/3) × (2/5) × (1/4)
Simplifying this expression:
Expected number of defective items = 6/840
Therefore, the expected number of defective items in the selected 4 items from the shipment is 6/840, which can be further simplified if needed.