Thirty percent of the customers at a hardware store pay using a credit card. Con

Question

Thirty percent of the customers at a hardware store pay using a credit card. Consider the following two probabilities:

Probability I – When 100 customers are selected at random – the probability that between 20 and 40 of them will use a credit card.

Probability II – When 10 customers are selected at random – the probability that between 2 and 4 of them will use a credit card.

Which is true? Pick one and explain briefly.

A.   The probability described in I is larger.

B.   The probability described in II is larger.

C.   The probabilities described in I and II are the same.

Explanation / Answer

Probablity II is larger suppose PII =p

case 1 can be formed by following combinations of case2 as:

for every group of ten 2 to 4 people pay through credit card i.e. P=p10

in one group 4-8 people pay through credit card and apart from rest except one ,in remaining group number are same 2-4 people, therefore for this case Probablity= 10C1p' 9C8 p p' is probablity of chosing 4-8 people from one group and for 2-4 p is same.

similarily all scenarios are formed ....

And P I is summition of all scenarios.

as you note that p' is decreasing very rapidly for others whose number are more than 4-6 or 6-8 or so on. .. as average is 30 percent and you pick 6-8 from group of then then probablity is very very low...

So P I =p10 + 10C1p' 9C8 p +.......

as p is less than 1 so p10 is very very less and other than p whose number are greater than 2-4 p' for them are too low. so as a result P I isless than P II

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