a box contains 20 fuses ,out of which ,5 are defective.if a sample of 3 fuses is chosen from the box at random without replacement,find the probability that x fuses in this sample will be defective
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To find the probability of selecting a certain number of defective fuses from a sample of three, we will use the concept of combinations.
First, let’s determine the total number of possible combinations of three fuses that can be chosen from a box containing 20 fuses. This can be calculated using the formula for combinations:
nCr = n! / (r! * (n - r)!)
where n is the total number of fuses and r is the number of fuses chosen.
nCr = 20! / (3! * (20 - 3)!)
= 20! / (3! * 17!)
Simplifying this expression:
20! = 20 * 19 * 18 * 17!
3! = 3 * 2 * 1
we get:
nCr = (20 * 19 * 18 * 17!) / (3 * 2 * 1 * 17!)
= (20 * 19 * 18) / (3 * 2)
= 1140
Therefore, there are 1140 possible combinations of three fuses that can be chosen from the box.
Next, we want to find the probability of selecting x defective fuses from the sample of three.
To select x defective fuses, we need to choose x fuses from the 5 defective fuses and (3 - x) fuses from the remaining (20 - 5) non-defective fuses. This can be calculated using the same combination formula:
nCr = n! / (r! * (n - r)!)
For x defective fuses, we have:
- r = x (number of defective fuses chosen)
- n = 5 (total number of defective fuses)
Therefore, the number of combinations of x defective fuses chosen from a set of 5 defective fuses can be calculated as:
5Cx = 5! / (x! * (5 - x)!)
Simplifying this expression:
5! = 5 * 4 * 3 * 2 * 1
x! = x * (x - 1) * (x - 2) * ... * 3 * 2 * 1
(5 - x)! = (5 - x) * (4 - x) * (3 - x) * ... * 3 * 2 * 1
Therefore, the probability of selecting x defective fuses from the sample can be calculated as:
P(x defective fuses) = (5Cx * (20 - 5)C(3 - x)) / 1140
Let's substitute values for x and calculate the probabilities:
For x = 0:
P(0 defective) = (5C0 * 15C3) / 1140
For x = 1:
P(1 defective) = (5C1 * 15C2) / 1140
For x = 2:
P(2 defective) = (5C2 * 15C1) / 1140
For x = 3:
P(3 defective) = (5C3 * 15C0) / 1140
Calculating these probabilities will give you the desired probabilities of selecting x defective fuses from the sample of three.