From a lot of 10 items containing 3 defectives, a sample of 4 items id drawn at random. Let the random variable X denote the number of defective items in the sample. If the sample is drawn randomly, find (i) the probability distribution of X (ii) `P(Xlt=1)` (iii) `P(X<1)` (iv) `P(0

Explanation

Given, from a lot of `10` itmes, `3` are defective

Good items `=10−3=7`

Let `x` represents the number of defective items

Clearly values of `X` are `0,1,2,3`

`P(x=0)=P(GGGG)`

`=p( text(good items))`
`=7/10xx6/9xx5/8xx4/7=1/6`
`P(x=1)=P(text(three good and one defective))`

`=P(BGGG)+P(GBGG)+P(GGBG)+P(GGGB)`
`=3/10xx7/9xx6/8xx5/7+7/10xx3/9xx6/8xx5/7+7/10xx6/9xx3/8xx5/7+7/10xx6/9xx5/8xx3/7`
`=1/8+1/8+1/8+1/8`
`=4/8=1/2`
`P(X=2)=P(text(two good and two defective))`
`=P(BBGG)+P(BGGB)+P(GBBG)`
`=3/10xx2/9xx1/8xx7/7+3/10xx7/9xx1/7+3/10xx2/9xx7/8xx1/7+7/10xx3/9xx2/8xx1/7`
`=1/120+1/120+1/120+1/120`
`=4/120=1/30`
Probability distribution is following:
(i)
`X 0 1 2 3 `
`P(X) 1/6 1/2 3/10 1/30`
`(ii) P(xle1)=2/3`
`(iii) P(x<1)=P(X=0)=1/6`
`(iv)P(0ltXlt2)=P(X=1)=1/2`