In our Winter/Spring 2022 newsletter, we shared one of Dan’s Challenge Problems to BEAM Pathway students:
Crystal is filling in the blanks in the following addition problem:
_ _
+_ _
____
_ _ _
First, she chooses a random two-digit number for the first line. Then, she chooses a random two-digit number for the second line. Finally, she chooses a random three-digit number for the third line. What is the probability that the addition comes out correct?
Remember: Numbers don't start with 0.
Let’s start!
To figure out the probability, we need to figure out the total number of ways to fill in two two-digit numbers and one three-digit number, and the total number of ways to fill in the addition correctly. Then the probability is
(# of ways to fill in correctly)
(total # of ways to fill in)
The total number of ways to fill it in is not too hard. There are 90 two-digit numbers. You can figure that out either because you have all the options from 10 to 99, or you can think of it as 9 options for the first digit (because you can't use zero), times 10 options for the second digit: 9*10 = 90. There are 900 three-digit numbers (going from 100 to 999, or 9*10*10). So in total, you have 90 options for the first two-digit number, times 90 options for the second, times 900 options for the three-digit number: 90*90*900 = 7,290,000.
Now we need to find the total number of ways to fill it in correctly. Here are two ways to find this total.
Solution 1:
Suppose we fill in a first two-digit number. Then to find an addition that works, all we need to do is fill in a second two-digit number so that the total is a three-digit number. The only thing that could go wrong is the sum might be too small: it might be less than 100. So which ones do work? Well...
If the first two-digit number is 10, then the second two-digit number can be 90-99 to get a three-digit number as the answer. That's 10 possibilities.
If the first two-digit number is 11, then 89-99 works, giving 11 possibilities.
If the first two-digit number is 12, then 88-99 works, giving 12 possibilities.
And so on . . .
Until, if the first two-digit number is 90, then every single other two digit number 10-99 works, giving 90 possibilities.
It's the same for 91, 92, all the way up through 99: each one gives 90 possibilities because any two-digit number you add gives a three-digit number.
So there are 10 ways to make the addition work if your first two-digit number is 10. There are 11 ways to make it work if your first two-digit number is 11. The total number of ways to make it work is:
10 + 11 + 12 + ... + 89 + 10*90.
The 10*90 comes up because for all ten numbers 90-99, all 90 two-digit numbers make it work.
How do we add these numbers? Well, you can use the "rainbow method" you might have learned sometime, or you can remember the formula
1 + 2 + 3 + ... + n = n(n+1)/2.
So...
1 + 2 + 3 + ... + 89 = 89(90)/2 = 4005.
1 + 2 + 3 + ... + 9 = 9(10)/2 = 45.
And then we subtract these to get 10 + 11 + 12 + ... + 89 = 4005 - 45 = 3960.
Add that to 10*90 = 900 and we get 4860 ways to correctly fill in two two-digit numbers and a three-digit number.
The probability is then 4860/7,290,000 = 0.06667%. That translates to 1/1500, so about one out of every 1500 times, Crystal will randomly fill in the addition correctly!
Solution 2
There's another way to figure out the total number of ways to fill in the addition correctly. Consider options for the three-digit number:
If it's 199 or bigger, it doesn't work, because the largest possible number we can get is 99 + 99 = 198.
If it's 198, there is exactly one way to fill in the 2-digit numbers: 99 + 99.
If it's 197, there are exactly two ways: 99 + 98 or 98 + 99.
If it's 196, there are exactly three ways: 99 + 97, 98 + 98, or 97 + 99.
If it's 195, there are exactly four ways: 99 + 96, 98 + 97, 97 + 98, or 96 + 99.
And so on . . .
Until with 109, there are 90 ways to do it: 99 + 10, 98 + 11, 97 + 12, ... , 10 + 99.
For 108, there are 89 ways to do it: 99 + 11, 98 + 12, ..., 11 + 99
And so on . . .
Until 100, where there are 81 ways to do it: 90 + 10, 89 + 11, ... , 10 + 90
Below 100, it is no longer a three-digit number.
So to find all the ways that work, the final step is to add up all of the ways we've found so far. That means we must add
1 + 2 + 3 + 4 + ... + 90 + 89 + 88 + ... + 81.
There are lots of ways to do this addition. We could use the "rainbow method" or follow this formula:
1 + 2 + 3 + ... + n = n(n+1)/2.
So we can add 1 + 2 + 3 + ... + 90 like this:
90(91)/2 = 4095.
We still need to add 89 + 88 + ... + 81. To do this, we can add 1 + 2 + ... + 89 = 89(90)/2, and subtract 1 + 2 + 3 + ... + 80 = 80(81)/2. That gives 4005 - 3240 = 765.
So in total, there are 4095 + 765 = 4860 ways to correctly fill in the addition. We got the same result as before, which again gives approximately a 1/1500 probability.