0
$\begingroup$

Such a straightforward question: of the digits 1,2,3, 7, 8, what is the probability of "making" a 3 digit number greater than 700?

For my sample space I think of all the ways of selecting 3 digits from 5. $5C3=10$. In order to be greater than 700 I must select either a 7 OR 8 first.

So, I think, how many different ways can I select a 7 first = just 1. How many different ways can I select an 8 first = just 1.

So, would seem to me the answer would be 2/10=1/5. Book says 2/5. Can you show me where my logic is incorrect?

I understand that there is a way to look at this using permutations for the sample space, but I thought I could do this with combinations.

  • 0
    Are you allowed to reuse digits? That would seem to lead to the book answer – 2017-01-19

2 Answers 2

0

You have ${5\choose3}$ ways of selecting $3$ digits, but that is not the amount of different numbers you can make i.e. that is not the size of the sample space, those are just groups of digits. The problem is, the same group can generate numbers that are, or are not, greater than $700$. For example, the group

$$\{1,2,7\}$$ can create both $721$ and $127$.

So you have two ways of going about it:

  • realize that there are $5!$ total different numbers, from which $2\cdot4!$ start with $7$ or $8$, making $\frac{2\cdot4!}{5!} = \frac25$;

  • realize that all it matters is picking the first digit right. You have $5$ possible ones and only $2$ will do, making $\frac25$.

  • 0
    Sticking with the sample spce discussion, I understand that the sample space has some successes and some failures, but I thought that was the definition sample space. – 2017-01-19
  • 0
    Yes; your sample space is composed of all the numbers that have digits in the set you wrote. But out of those, you want to select only the ones that are successes. That is how you count probabilities. You divide the number of successes in the sample space by the size of the sample space. The size of the sample space is $5!$ and not $5\choose{3}$ – 2017-01-19
0

For my sample space I think of all the ways of selecting 3 digits from 5.$ 5C3=10$.

That counts the ways to select three digits, but not how to arrange them in a specific order. Additionally, you aren't considering the possibility of using the same digit multiple times.

In order to be greater than 700 I must select either a 7 OR 8 first.

So, I think, how many different ways can I select a 7 first = just 1. How many different ways can I select an 8 first = just 1.

To be consistent with the sample space. of selecting three digits in three places, you should count ways to select 7 or 8 for the first place, and two digits for the remainder.

But let's go back a step.

In order to be greater than 700 I must select either a 7 OR 8 first.

Indeed.   So just focus on that.   You do not need to consider the entire sample space.

What is the probability of selecting those two digits for the first place, when you are selecting from five digits for that place?