Which Probability Calculation is Correct for Defective Bulbs in a Sample?

[zorro]

Homework Statement

In a box containing 100 bulbs, 10 are defective. The probability that out of a
sample of 5 bulbs, none is defective is?

The Attempt at a Solution

I see two approaches to this problem.

1) Out of 90 non-defective bulbs, we can chose 5 in ⁹⁰C₅ ways.
There are a total no. of ¹⁰⁰C₅ ways.

So required probability =⁹⁰C₅/¹⁰⁰C₅=0.5838

2) Out of the sample of 5 bulbs,
Probability of a non-defective bulb = 90/100=9/10
In 5 bulbs, required probability = (9/10)⁵=0.59049

The two answers don't differ by much. Which one is correct and why?

 

[LeonhardEuler]

2) Out of the sample of 5 bulbs,
Probability of a non-defective bulb = 90/100=9/10
In 5 bulbs, required probability = (9/10)⁵=0.59049

You can multiply the probabilities of independent events to calculate the probability of all of them occurring together. Is pulling a second non-defective light bulb independent of pulling a first one?

In other words, suppose you are picking the second lightbulb from the box. Does the probability depend on the condition of the first lightbulb?

 

[zorro]

In the selected sample, you don't take away the bulbs after picking them. In other words, the bulbs are replaced in the sample. So they are independent events.

 

[LeonhardEuler]

So when they say "a sample of 5 bulbs", you can have the same bulb multiple times in the sample?

 

[HallsofIvy]

Homework Statement

In a box containing 100 bulbs, 10 are defective. The probability that out of a
sample of 5 bulbs, none is defective is?

The Attempt at a Solution

I see two approaches to this problem.

1) Out of 90 non-defective bulbs, we can chose 5 in ⁹⁰C₅ ways.
There are a total no. of ¹⁰⁰C₅ ways.

So required probability =⁹⁰C₅/¹⁰⁰C₅=0.5838

2) Out of the sample of 5 bulbs,
Probability of a non-defective bulb = 90/100=9/10
In 5 bulbs, required probability = (9/10)⁵=0.59049

The two answers don't differ by much. Which one is correct and why?

This is selection without replacement. In (2) you are calculating the probability with replacement. That is, as if you take a bulb, test it, put it back in the box and choose again, with a (slight) chance of getting the same bulb again.

Instead you could argue that, at first, there are 90 non-defective bulbs out of 100 so the chance that the first bulb selected is non-defective is 90/100= .9. But then there are 89 non-defective bulbs left among 99 bulbs. The chance of selecting a non-defective bulb the second time is 89/99, not 90/100 again.

The probability of selecting 5 non-defective bulbs is (90/100)(89/99)(88/98)(87/97)(85/96)= 0.5838 as in (1).

 

[zorro]

This is selection without replacement. In (2) you are calculating the probability with replacement. That is, as if you take a bulb, test it, put it back in the box and choose again, with a (slight) chance of getting the same bulb again.

Instead you could argue that, at first, there are 90 non-defective bulbs out of 100 so the chance that the first bulb selected is non-defective is 90/100= .9. But then there are 89 non-defective bulbs left among 99 bulbs. The chance of selecting a non-defective bulb the second time is 89/99, not 90/100 again.

The probability of selecting 5 non-defective bulbs is (90/100)(89/99)(88/98)(87/97)(85/96)= 0.5838 as in (1).

I understood your explanation.
But the thing is that, in my text the answer given is (0.9)^5. Not only this, I searched this question in google books and found that everywhere the answer is same as above.

ex:

18. The probability of getting a defective bulb from the box, is 1 / 10.

Hence using binomial distribution,

the required probability , is (0.9)^5.

Hence the option is C.

Source - http://creatorstouchglobal.com/gm/index.php?option=com_content&view=article&id=63&Itemid=121

 

[LeonhardEuler]

It is really seems wrong on that website. If the question was "What is the probability of pulling a non-defective bulb 5 times if the bulbs are replaced after each drawing" then the answer would be (0.9)^5, but "a sample of 5 bulbs" strongly implies no replacement.

 

[zorro]

hmmm...its possible that the answer is wrong.