Question

A manufacturer received an order of 210 computer chips. Unfortunately, 10 of the chips are defected. To test the shipment, a quality-control engineer randomly selects 24 chips from the box of 210 and tests them. The random variable X represents the number of defective chips in the sample.

What is the probability of obtaining 5 defective chips?
What is the probability of obtaining 3 defective chips?
What is the probability that the QC engineer will not find any defective chips?
Probability of finding 13 defective chips?
How many defective chips would you expect to select?

Answer 1

@jim_thompson5910 last one i need help with!

Answer 2

well right off the bat, I can answer "Probability of finding 13 defective chips?" without having to do any math at all really

do you see what I mean? or no?

Answer 3

yeah that one is 0 right?

Answer 4

since theres only 12?

Answer 5

good, it's impossible since there are only 10 defective chips

Answer 6

or 10 i mean

Answer 7

i've tried everything to find the others and im stumped :/

Answer 8

let me think on those

Answer 9

no problem!

Answer 10

what kind of calculator do you have?

Answer 11

TI-84

Answer 12

ok we'll have to rely on the nCr function

do you know how to find that?

Answer 13

yes!

Answer 14

have you learned about the hypergeometric distribution?

Answer 15

no :/

Answer 16

That distribution relies on the nCr formula

Answer 17

we pick a sample of 24 chips

Answer 18

in this sample, we want to know the probability of having exactly 5 bad chips

Answer 19

There are 10 bad chips overall

So there are 10 C 5 ways to pick 5 bad chips. We won't calculate 10 C 5 since we'll let the calculator do this for us

Answer 20

so i got 252?

Answer 21

our sample size is 24

if we allot 5 spaces for the bad chips, we have 24 - 5 = 19 spaces for good chips

Answer 22

There are 210 - 10 = 200 good chips

so there are 200 C 19 ways to pick 19 good chips (from a pool of 200)

if you were to calculate 200 C 19 you'll get a very large number

Answer 23

put together, there are (10 C 5)*(200 C 19) ways to pick 5 bad chips AND 19 good chips

Answer 24

this is out of 214 C 24 ways to pick 24 chips total

Answer 25

so in your calculator, you will type this in

( (10 nCr 5)*(200 nCr 19) )/( 214 nCr 24 )

Answer 26

i got .00125

Answer 27

oh sry, not 214, 210

Answer 28

fixed

( (10 nCr 5)*(200 nCr 19) )/( 210 nCr 24 )

Answer 29

oh okay now i got .002019

Answer 30

here's what I get

Answer 31

and you got the same

Answer 32

is the next the same just substitute 3 for 5?

Answer 33

for the next part, you are changing the "5 bad chips" to "3 bad chips"

so you'll have

( (10 nCr 3)*(200 nCr 21) )/( 210 nCr 24 )

Answer 34

the 5 changed to 3
the 19 changed to 21

Answer 35

also I recommend you read this page

http://staffwww.fullcoll.edu/kchang/MATH%20120%20Primavera%202012/6.4_HypergeometricProbability.pdf

as it has some other examples on the hypergeometric distribution

Answer 36

.0746?

Answer 37

getting the same

Answer 38

thanks i will!

Answer 39

for part c, you'll have 0 defective chips in the sample

so you'll have

( (10 nCr 0)*(200 nCr 24) )/( 210 nCr 24 )

Answer 40

.28875?

Answer 41

same

Answer 42

part d is already done

Answer 43

sweet! just one more :)

Answer 44

part e will require the formula in the blue box on page 5

http://staffwww.fullcoll.edu/kchang/MATH%20120%20Primavera%202012/6.4_HypergeometricProbability.pdf

Answer 45

the first formula in that box

Answer 46

okay i see it

Answer 47

n = 24 (sample size)
k = 10 (number of defective chips...ie number of "successes")
N = 210 (overall population size)

Answer 48

.9842? @jim_thompson5910

Answer 49

nope

Answer 50

use the n*k/N formula

Answer 51

ah i was hoping i got it right after all that

Answer 52

oh 1.14?

Answer 53

@jim_thompson5910

Answer 54

n*k/N = 24*10/210 = 240/210 = 1.142857

so you are correct

Answer 55

thank you so much Jim! i know it was a lot but i appreciate it so much

Answer 56

sure thing