Finding the probability of "at least one"

Intro

Howdy!  I’m Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today were going to learn how to find the probability of at least one.  Here’s our problem statement: Subjects for the next presidential election poll are contacted using telephone numbers in which the last four digits are randomly selected with replacement.  Find the probability that for one such phone number the last four digits include at least one zero.

Solution

OK, so what we’re looking at here is the probability of at least one of those four digits the end of the phone number being zero.  And to do that, we actually remember that, whenever looking for the probability of at least one, that's one minus the probability of none.  So the probability that none of those last four numbers are zero subtracted from one is going to give us a probability of at least one.

So what’s the probability that none of them are zero?  Well, it’s the probability that the first number is not zero and the probability that the second number is not zero and the probability that the third number is not zero and the probability that the fourth number is not zero.

OK, so from this point what do we do?  Well, what’s the probability that the first one is not zero?  Well, we go through and substitute, we find that it's a part over the whole.  So there are nine numbers that are not zero, and there's 10 numbers total (0 through 9), so the part over the whole is 9/10.  This is the probability that the first number is not zero.  And of course is just statistics speak for multiplication, so that's what we have here.  And then we can continue on and find that it's the same thing for the rest of the numbers.  So the probability that the second one is not zero — you got 10 numbers to choose from, one of them is not zero, so there’s nine to choose from out of the 10 total.  And it just goes on like that, so we can just continue on like this.

Now, at this point in the expression, we can simplify this.  We got nine tenths multiplied by itself.  There’s four of them, so I can actually just simplify that by saying nine tenths to the fourth power, which, when I punch that out on my calculator, I get 0.6561.  Subtract that from 1, and I get 0.3439.  I’m asked to round to three decimal places.  Nice work!

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