Expert: Paul Klarreich - 3/26/2006

Question
A tennis match is played where the winner must win 2 sets to win the match. (Think of the outcomes in terms of a tree diagram).

   Number of possible outcomes =

I thought the answer was 3 possible outcomes being win 2
loose 2
win 1, loose 1

however this answer is not right

Answer
Hi, Kevin,
You wrote:
Subject:  Probability/ Combination
Question:  A tennis match is played where the winner must win 2 sets to win the match. (Think of the outcomes in terms of a tree diagram).

  Number of possible outcomes =

I thought the answer was 3 possible outcomes being win 2
loose 2 win 1, loose 1

>> That's LOSE, please.  Use your head, not your spelling checker.

however this answer is not right
---------------------------------------
Try this approach:

The two play a first set.  There are two possible outcomes -- Kim wins, Maria wins.  FOR EACH of those, construct the next two branches:  Again, the two possible outcomes are -- Kim wins, Maria wins.  So we now have four branches.  (BTW, each of these branches has probability 0.25)

Continue to a third set, WHICH MIGHT NOT ACTUALLY BE PLAYED.  We get to four branches, each of which now has probability 0.125, WHETHER IT GETS PLAYED OR NOT.  You can take the position that if Kim won the first two sets, they play a third set for practice, but the spectators all left to watch another match.

The diagram would look like this:

WARNING: THE MATERIAL BELOW MAY CONTAIN FRACTIONS AND OTHER MATERIAL INAPPROPRIATE FOR CERTAIN COMPUTING SYSTEMS.  BE SURE TO VIEW IT IN A FIXED-SIZE FONT, SUCH AS COURIER.


Set 1:            K (0.5)             M (0.5)
                 |                 |
           -------------       -------------
           |           |       |           |
Set 2:    K(0.25)    M(0.25)  K(0.25)    M(0.25)
           |           |       |           |
       ---------   -------- --------    --------  
       |       |   |      | |      |    |      |
Set3:   K*     M*   K      M K      M    K*     M*

In the last line, each entry has probability 0.125 and the outcomes marked with stars are played 'for practice'.

To find the probability of an EVENT, which is not the same as an OUTCOME, (remember, an event is a set of outcomes) just find the lines (complete branches) that correspond to the event and add their total probabilities.

Such as:

p(Kim wins the match) = sum of probabilities for the lines:
KKK, KKM, KMK, MKK = 0.5  as it should be.

OR:

p(match requires three sets) = sum for KMK, KMM, MKM, MKK = 0.5