Question

2 bulbs out of a sample of 10 bulbs manufactured by a company are defective. The probability that 3 out of 4 bulbs bought by a customer will not be defective is

Comments

i was trying to solve it using binomial formula but got wrong answer,why cant we solve it using binomial formula??

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Is it 0.4096?

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no,it is 8/15..

Answer 1

For Binomial distribution to be used , these conditions should be followed :

a) In such trial we have either success or failure ..

b) Sampling should be from infinite population(here it is referred to no of bulbs)

c) Sampling from finite population but with replacement

So here we cannot use binomial distribution as none of b) and c) is satisfied..As no of bulbs is finite and nothing is given so we consider without replacement in such scenario by default..

So we use hypergeometric distribution here..Let me give an idea about this

We have 8 non defective bulbs ..But in outcome we require 3 such bulbs..So this can be done in ⁸C₃  ways..And similarly for defective bulbs we have ²C₁  ways..

So favorable outcomes of choosing desired bulbs =  ⁸C₃  * ²C₁  ways

And we can choose 4 bulbs out of 10 bulbs in ¹⁰C₄ ways which is the total outcome number..

Hence P(3 non defective out of 4 bulbs chosen)  =  (⁸C₃  * ²C₁) / ¹⁰C₄

                                                                                               =  8 / 15

Hence 8 / 15 is the correct answer..

Comments

i was also getting this answer but answer given is 8/15

and here is the method(which also seems to be correct)

Explanation:

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They didn't say that there are only 10 bulbs to buy from. So 0.4096 is correct.

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Plz check my answer @akashsheoran..

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@habib khan,i did not get the conditions for binomial theorem.

the first condition which is of success and failure is met here..as being defective is failure and non defective is success.

and i did not get last two conditions

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Either the number of bulbs will not be mentioned meaning infinitely many or if given it has to be mentioned explicitly bulbs are chosen with replacement ..Then it will work for finite number of bulbs as is the case here..

Else the method which I have to be mentioned has to be used..As nothing is mentioned , so we assume bulbs are chosen without replacement which is the default assumption.

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if bulbs will be infinite then what will we take in place of 'n' in the formula??

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Here n will be number of trials rt as given to be 4 here..But population is different..Infinite population may be represented by giving just proability of defective bulb , say 0.2 instead of saying 2 bulbs out of 10 bulbs are defective..

And moreover in each trial the probability of success will not change in Binomial Distribution..But since we have considered here without replacement , so hypergeometric distribution is used instead..

I hope u get it now..

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yes,i have understood the concept,but  by this line "2 bulbs out of a sample of 10 bulbs manufactured by a company are defective."we can also think that they have given the probability in this way instead of directly writing 0.2 and it is written that 'sample' of 10 bulbs is taken,so sample must be taken from a large lot.

i dun know whether i have made my point clear or not..

just clear one thing,if in the question it was given that prob of being defective is 0.2 instead of saying that 2 out of 10 bulbs are defective then we colud have solved it using binomial distribution..right??

thanks a lot

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@Habib. In binomial theorem, the positions of samples are important, right? Here, there is no suh constriant. Hence, we didnt consider binomial distri.

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But probability of each trial changes as by default we take without replacement i.e. if a bulb is drawn then it will not be reconsidered for further trials..

Also total number of bulbs is 10 which is finite here..

Hence u cannot use binomial distribution here.. Here u have to use hypergeometric distribution..

Plz check about this term first.. U will get it

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@habib. you are right. Anyways, finiteness isnt the condition for not using Binomial distributuion.

Thanks :)

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@Sushant , So is it just like if we pick a object from sample with replacement then binomial else default is hypergeometric ? finiteness is not a condition ?

I think that finiteness should also be the condition because here in this if they would have not given finite number of bulbs (that is 10) then we couldn't have done 8C3 as we dont know the total and we would be forced to apply binomial distribution.