2008 has 2 digits that are the same -- A Probability Investigation For Middle Schoolers And Beyond

As the school year is beginning...

Which would you conjecture is more likely:
No digits the same in a 3-digit number or no digits the same in a 2-digit number?

You have 30 seconds to choose one of these - - - - - - - - - -
NOW WRITE YOUR GUESS ON YOUR PAPER and compare with your partner. Take one minute to discuss your thoughts...

Alright, I know some of you take exception to wasting these 30 seconds. What could be gained from such 'blind' guessing without the time to really think it through and work it out. I often used device this to encourage youngsters to react instinctively and to learn to trust their intuition. How many times have all of us had the experience of not trusting ourselves, only to find later that we were right. If it turns out that this gut reaction is not supported by the data, then the mathematical researcher (or the experimental mathematician in this case) revises the hypothesis. Ultimately, one attempts to validate one's conjectures via logic (deduction, induction, etc.). If you're still not convinced this is worthwhile, it's only a suggestion...

Now we're past the prelims. Our goal is to have our students begin with solving a particular case of the problem above and then to develop a general relationship for:
The probability that an N-digit positive integer will have N different digits. Of course, N is restricted to be in the range 1..10. We would hope our students from middle school on would recognize that the probability for N = 1 is 100%, whereas the probability for N = 11,12,13,... would be zero! Yes, we would hope!

(1) Show that the probability a 2-digit positive integer has different digits is 90%.
Comments: This is a well-known and fairly basic problem, but this is just the jumping-off point for this investigation. Various methods are likely here, depending on the background of the student. The middle school student (and many secondary as well) would likely list or count the number of 2-digit numbers with different digits. Some would realize that it might be easier to count those with identical digits and subtract from the total. More advanced students may use more sophisticated approaches for this and the other parts below. One could use this activity to develop the multiplication principle, permutations, use of factorials, etc. However, there is much to be gained from 'first principles.' Careful counting and making an organized list never go out of style!

(2) Show that the probability a 3-digit positive integer has 3 different digits is 72%.

(3) Complete the following table up to N = 10:
Note: P(N...) denotes the probability of the indicated outcome.

Number of Digits N.....P(N different digits)

...........1......................100% or 1
...........2..................... 90% or 0.9
...........3......................72% or 0.72
...........4......................50.4% or 0.504......
.
.
..........10.....................................................


(4) Time to revisit your original conjecture.... Explain why the probabilities decrease as the number of digits increase.
Note: One could give a purely descriptive explanation here.

(5) For more advanced groups:
Develop a formula for P(N).

(6) For more advanced groups:
Enter your expression from (4) into Y₁ of the Y= menu in your graphing calculator. Set up a TABLE with Start value of 1, increment (Δ) = 1 and Auto for Indpnt and Depend. Display your table and check the values you found from your own table.

(7) [Optional]
Closure: Write 3 ideas, methods, strategies, mathematical principles, etc., you have learned from this activity.