Probability of Matching Times on a Clock

Date: 09/14/2004 at 12:17:07
From: Michael
Subject: What is the probablity

What is the probability of two different times within the same hour 
ending in the same last digit? Like 08:13 and 08:43?

I don't know the process of calculating the probability.

Date: 09/14/2004 at 12:59:58
From: Doctor Edwin
Subject: Re: What is the probablity

Hi, Michael.

In general, the chance of an event occurring is

         # of ways the event could happen
      --------------------------------------
       total # of ways things could turn out

So for example my chance of rolling a one or a two on a six-sided die 
is:

           # of ways I could roll 1 or 2 
      ---------------------------------------
       total # of ways the die could come up

or:

           2
          ---
           6

which is just 1/3.

Your question has an interesting twist.  The most obvious answer is
1/10 or 10%.  Let's ignore everything but the last digit.  If I pick a
number between zero and nine, and you do the same, the chance that you
picked the same number I did is 1/10.  So if I pick a time at random,
and you pick a time at random, the chance that the last digits will
match is still 1/10.

But in your problem, it adds two additional pieces of information. 
First, it says that the times must be different. So we can't both 
pick 8:13, for example.  The other piece is that the two times must 
fall within the same hour.

In order to figure out the probability of your time event, we have to 
figure out how many ways we could have the same number at the end, 
while picking DIFFERENT times in the same hour.

Suppose like in your example I pick 8:13. Now, how many ways can you 
pick a time that ends with the same digit as mine?

8:03
8:13
8:23
8:33
8:43
8:53

but one of those is the same time I picked, and you're not allowed to 
pick that one, so you're down to 5 possible ways to pick the time.

So your probability is

                                5
             ----------------------------------------
              total # of times you could have picked

So how many times could you have picked? There are 60 minutes in an 
hour, and you're not allowed to pick one of them. Can you figure out 
the probability from there? Write back if you're stuck.

- Doctor Edwin, The Math Forum
  http://mathforum.org/dr.math/