[RESOLVED] A "simple" Probability Question

[Code Doc]

I have a list of 26 names. I'm selecting names from the list at random. Only one of the names from the 26 can answer a question correctly. As I select and miss, I remove a faulty name from the list, so it shrinks as I go along.

What is the probability that I make 25 straight improper selections so that only the last name that I select on the list answers the question correctly?

LOML blurted out, "It's about 0.04." How accurate is her analysis?

[TBeck]

I think this is pretty accurate,

the first time you have 25/26 chance of picking an incorrect answerer
then a 24/25 chance, then 23/24, and so on, so the product of these gives you:

25/26 * 24/25 * 23/24 * ... * 2/3 * 1/2 * 1/1 (where the last 1/1 is for the person who is answering correctly)

simplifying this we obtain

25!/26! * 1 = 25!/26! = 0.038461538461538461538461538461538

[Code Doc]

Here's some VB code I wrote:

DefInt A-Z
Private Sub Form_Load()
ListSize = 26
Odds! = 1 ' Initialize
For I = ListSize To 2 Step -1
    Odds! = Odds! * (I - 1) / I
Next
Label1.Caption = "Probability (missed 25 straight) = " + Format$(Odds!, "0.#######")
End Sub

Same answer. 0.3846154. Thanks a bunch, TBeck. Now, how did LOML do all this in her head?

[TBeck]

well there is a little trick to doing this factorial calculation in your head

25!/26! = (25*24*23*...*3*2*1) / (26*25*24*23*...*3*2*1)

which can be reduced to 1/26

so if you can compute 1/26 in your head you can know it is about 0.038 but I think that she may have taken 1/26, said it was about 1/25 which is exactly 0.04

[Code Doc]

For some weird reason, the 1/26 probablilty is not intuitively obvious to me. When I first thought about the problem, I estimated the odds to be much slimmer than that.

It appears that missing 25 in a row from a shrinking list is the same as randomly guessing the correct name on the very first attempt.

This problem also reminds me of the matching tests we used to take in school. As you continued to match correct answers, the list size shrank and it became easier to get the correct answer on subsequent attempts--provided you made no mistakes as you went along.

Guesssing all the answers correctly on a 26-item matching test is a virtual impossibility. (But that's another problem.)