The Professor's Response

Hi Professor,

For Now You Try It! 3.8 Advanced Probability I don't understand number 3. I am confused between what is meant by the numbering of the marble compared to the color of the marble and how to put it in the combination formula.

A box of numbered marbles has 12 red, 12 blue, 12 green, and 12 yellow marbles. The

marbles for each color are numbered from 1 to 12. That is there is a unique number on

each marble, so no marble is exactly the same as any other marble in the box. When

reaching into the box to randomly draw five marbles without replacement, what is the

probability of getting exactly four marbles of the same color (note: the numbers matter

but the order does not)?

Also, I wanted to know when I took the Sample Test for number 3 you pointed out that it is without replacement because the probability is 3/120 which is less than 5%. On the test would you point that out or do would we have to figure it out by dividing the sample by the population?

3. A bin at a local big box store has 120 discounted packs of batteries of which 9 are old and contain batteries that no longer work. If three of the packs of batteries are randomly selected for purchase, what is the probability that all of the packs of batteries are good ones (treat this problem as if it involves selection with replacement since 3/120 is less than 5%)?

Thank you.

See the professor's answer below.

Hi Chantal,

I have posted a video below to explain question number 3 from the "Now You Try It!" section of 3.8.

As for the question on the sample test, don't worry about using that 5% rule on the exam. If the problem appears to involve a set of dependent events, just treat it as dependent. While the 5% rule allows us to avoid having to reduce the numerator and the denominator of the fraction, it is optional. We certainly will never go wrong treating the problem as if it were dependent (i.e. selecting without replacement) when it is in fact dependent. The main use of the 5% rule is when we are given the probability as a decimal or a percent and we do not know the total population size. For example, if the problem said 3% of the population has the flu. What is the probability that four randomly selected people all have the flu? The answer would be .03^4. However, this problem technically deals with dependent events, but, there is no way to treat it as dependent without knowing the total number of people in the population because you'd have to convert the 3% into a fraction (not 3/100 if that is what you were thinking, it would be something like 9,000,000/300,000,000) so you could reduce the numerator and denominator by one as you went. In this case, the 5% rule makes that approach unnecessary. So, in short, don't even consider using the 5% rule unless the problem gives you the probability as a decimal or a percent.

Hope that helps,

Professor

Hi Chantal,

I think you have misunderstood the 5% rule. The 5% does not refer to the probability that is used in the problem. It refers to the number of subjects or items you are selecting in the problem. In the problem above, you are selecting four 18 - 36 yr olds. Since four people is much less than 5% of the total population of 18 - 36 yr olds, the 5% rule can be used. The rule allows us to treat the selections as if they are with replacement.

Sincerely,

Professor McGuckian